Collateral Risk, Repo Rollover and Shadow Banking

Shengxing Zhang

ú†

Department of Economics, New York University

[download the latest version]

January 26, 2014

Abstract

I build a dynamic model of the shadow banking system and the interbank repo market to

understand their e�ciency and stability. The model emphasizes a key friction: the maturity

mismatch between short-term repo and long-term investments that banks need to finance. The

haircut, interest rate, and default rate of the repo contract are endogenously determined in the

model, as are the volume of lending and liquidity hoarding. Default is shown to be contagious.

Finally, when collateral risk increases unexpectedly, the haircut and interest rate overshoot,

triggering massive defaults and persistently hiking the default rate and depressing investment.

úemail: shengxing.zhang@nyu.edu†I am indebted to Ricardo Lagos, Douglas Gale, Boyan Jovanovic and Thomas Sargent for their support and

discussions. I am especially grateful to Ricardo Lagos and Douglas Gale for their guidance on the project. I also

thank David Andolfatto, Gara Afonso, Saki Bigio, Dan Cao, Katarína Borovicka, Jaroslav Borovicka, Emmanuel

Farhi, Klaus-Peter Hellwig, Ben Lester, Yaron Leitner, Jesse Perla, Edouard Schaal, Cecilia Parlatore Siritto, Chris

Tonetti, Laura Veldkamp, Venky Venkateswaran, Gianluca Violante and all other participants of the Sargent Research

Group, the Student Macro Lunch seminar of the Department of Economics at New York University, the Macro Lunch

seminar at NYU Stern, the student workshop at Wharton and the brown bag seminar at the Federal Reserve Bank

of St. Louis.

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1 Introduction

The shadow banking system is an essential part of the process of credit creation in modern banking.1

The process relies heavily on short-term debt instruments such as repurchase contracts (repo), a

short-term collateralized debt contract with safe harbor provisions.2 Just before the great recession,

the gross volume of outstanding repo contracts reached $10 trillion in both the US and Euro-zone

repo markets. To put this amount into perspective, it amounted about 70% of GDP in the respective

areas in 2007.3

In 2007, the risk on collateral assets increased unexpectedly due to the sharp decline in housing

prices.4 Concerned about the quality of collateral assets, financial intermediaries reduced their

repo exposure to each other and began hoarding liquidity. The haircut and interest rates shot up.

This eventually led to the downfall of Lehman Brothers, which ran out of resources to finance its

long-term investments.5

The crisis in the repo market and the shadow banking system exposed the instability of the

system and left us with the following questions. What is the source of systemic risk in the shadow

banking system? Are the e�ciency and stability of the system a�ected by frictions in the repo

market? What triggered the crisis? Why has the system not fully recovered even five years after

the outbreak of the crisis?

To answer these questions, I build a dynamic model of the shadow banking system and the

interbank repo market. Banks have an initial endowment of cash and collateral. Each bank

anticipates an investment option that arrives according to a Poisson process. While a bank waits

for its investment option to arrive, it lends through the repo market to banks that already have an1See Pozsar et al. [2013] for the credit intermediation process in the shadow banking system. And I will explain

its institutional features in more detail in the next section.2In legal terms, a repo contract is a combination of two outright transactions, sales at the moment the contract

is signed and purchase at a future date at a price according to the contract. Since it can be interpreted either as a

combination of two spot trades or as a secured loan, it helps some financial institutions circumvent legal restrictions

to lending to other institutions or to carrying out spot trade. Another di�erence between the repo contract and a

secured loan is that when a borrower defaults, the collateral asset is not subject to automatic stay. The safe harbor

provision makes financing through a repo contract popular. See Garbade [2006] for more details.3See Hördahl and King [2008].4ABX indices, price index for CDS over a collection of mortgage backed securities, dropped.5Gorton [2009] gives a detailed description of the unfolding of the events in the crisis.

2

investment opportunity. The model emphasizes a key friction: the maturity mismatch between the

short-term repo and the long-term investment that banks need to finance. The maturity mismatch

results in borrowers having to roll over their debt until the investment matures and they are able to

repay their loan. If the investment does not mature before the borrower reaches his (endogenous)

debt limit, the borrower will be forced to default.

The haircut, interest rate and default rate of the repo contract are all endogenously determined

in the model. As we will see, the endogenous haircut and default rate allow me to study how

changes in the primitives of the model a�ect the borrowing constraint and the externalities caused

by default.

I show that systemic risk arises because default triggers more default. When a borrower defaults,

the lender’s portfolio becomes less liquid: she gains a collateral asset but loses her claim to future

cash from her counterparty. Then, when the lender’s investment opportunity arrives, she relies

more on secured borrowing (backed by the collateral assets) and less on her own cash. This makes

her more likely to default because, other things being equal, she reaches her debt limit faster. Thus,

counterparty default is contagious.

To understand the failure of the shadow banking system and to check the robustness of the

theory, I extend the model to allow for collateral risk6 and find that an increase in collateral

risk increases counterparty default risk. When collateral risk increases, the repo market dries up

through two channels, liquidity hoarding and counterparty default. Banks reduce funding to the

repo market and hoard more liquidity to secure funding for their own investment in the future.

With less funding in the repo market, the equilibrium haircut and interest rate increase, the debt

limit is reached faster, and default is more likely. Counterparty default transforms the lenders’

portfolios and further decreases the supply of funding to the repo market. So, as collateral risk

increases, the e�ciency of the financial system declines.

I use the dynamic model to study both the e�ciency of the shadow banking system in the steady

state and the stability of the system in response to an unanticipated shock. The stability of the

system can be measured by two metrics: the magnitude of the initial response and the persistence6At the time, 50% of primary broker-dealers’ repo contracts are backed by such less-liquid securities as corporate

securities, mortgage-backed securities and other asset-backed securities, and 65% of the contracts are overnight. See

Adrian et al. [2009].

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of the e�ciency loss on the transition path to the new steady state. My objective is to understand

the stability of the financial system at the onset of the great recession. To do this, I characterize the

transition dynamics triggered by an unexpected increase in collateral risk. On impact, the hoarding

motive of lenders imposes a downward pressure on the supply of funds to the repo market. As a

result, both the haircut and interest rate increase sharply to clear the market. This leads to a

tightening of the borrowers’ debt limit and a massive default by those borrowers who suddenly

find themselves over their debt limit. In addition, the rest of the borrowers who started borrowing

before the crisis now face a debt overhang problem, which increases the default rate from that point

onward. Since counterparty default is contagious, the massive initial default and the increase in the

default rate have a persistent e�ect on the equilibrium path: Repo lending and investment remain

low, and the default rate remains high for an extended period of time. The systemic risk from

contagious counterparty default increases both the magnitude and the persistence of the e�ciency

loss on the dynamic transition path to a new steady state.

Literature review

Previous research has emphasized asymmetric information and market failure as causes of the

financial crisis (Chiu and Koeppl [2011], Camargo and Lester [2011]). Dang et al. [2009, 2012],

Gorton and Ordonez [2012], Farhi and Tirole [2012] and Hellwig and Zhang [2012] study the e�ect

of endogenous information structure and market liquidity. This paper takes a di�erent approach,

exploring the possibility that there is a simpler explanation for the crisis – namely, an increase in

collateral risk amplified through the mechanism of contagious counterparty default.

This approach is closely related to that of Kocherlakota [2001], in which collateral risk makes

it harder for a lender to enforce payment of the promised share of a project. As in Kocherlakota

[2001], it is too costly to collect the promised payment from borrowers, other than to seize the

risky collateral. The di�erence is that, when the value of collateral drops, lenders are not able to

withdraw funding from borrowers, even though only a tiny fraction of lenders observe the shock.

The repayment of repo contracts is supported mostly by debt rollover. But equilibrium rollover

will collapse when a small fraction of lenders want to withdraw, after observing the shock.

The term structure of repo borrowing in my model is exogenous. Brunnermeier and Oehmke

[2013] shows that the exemption from automatic stay of short-term repo contracts triggers a matu-

4

rity rat race, so, in equilibrium, banks borrow ine�ciently short-term. But it would be interesting

to endogenize the term structure in the future and study its e�ect, as in Williamson [2013].

The repo market in my model allows banks and investors to share profitable opportunities, as

in Kiyotaki and Moore [2002] and Berentsen et al. [2007]. Many authors have used the workhorse

model of Diamond and Dybvig [1983] to study traditional banking crises, but few have provided

models of the repo market. Martin et al. [2011] focuses on the repo market between cash providers

and financial intermediaries. I focus on repo lending between financial intermediaries to provide a

complementary approach to understanding systemic risk. Gennaioli et al. [2013] presents another

model of the shadow banking system, in the spirit of Diamond and Dybvig [1983]. In empirical work

by Gorton and Metrick [2012a], the authors take the view that the recent crisis was a system-wide

self-fulfilling bank run. While the Diamond-Dybvig model is important for understanding panics,

it is unclear how it explains the ‘runs’ that occurred in the bilateral repo market. Collateral plays

the same role in repo markets that deposit insurance plays in traditional banking and should have

prevented a self-fulfilling bank run in a Diamond-Dybvig-style model. I focus, instead, on studying

systemic risk and equilibrium dynamics in the repo market without runs.

This paper subscribes to the view, expressed in Sargent [2013], that liquidity problems are the

result of market incompleteness, and, hence, the solution to a liquidity problem is model-dependent.

I model the shadow banking sector with two questions from Moore [2011] in mind: Why do financial

intermediaries hold mutual gross positions, and do these gross positions create systemic risk?

The paper is related to the study of banks’ risk-taking behavior, as in Allen and Gale [2001]. In

my paper, investment in a profitable long-term project is risky, as it is uncertain when the project

will mature.

In my paper, a lender whose counterparty defaults is more likely to default when she invests in

her long-term project. The financial contagion e�ect in the paper is in the spirit of Allen and Gale

[2000]. Here, the e�ect takes place on the dynamic equilibrium path and has explicit implications

for variables such as the haircut.

Liquidity hoarding that results from collateral risk is reminiscent of the precautionary demand

for funding, as in Frenkel and Jovanovic [1980]. There is also a speculative motive for liquidity

hoarding – to buy assets at fire sale prices in the future – as in Gale and Yorulmazer [2013]. Gale

and Yorulmazer [2013] list two possible explanations for the phenomenon of liquidity hoarding:

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counterparty default risk and the fear that participation in lending may compromise a lender’s

future access to liquidity. Both ingredients contribute to the repo market freeze and to liquidity

hoarding in my model.

The (endogenous) characterization of haircuts is related to the study of endogenous leverage ini-

tiated by Geanakoplos and Zame [1997] and developed by Fostel and Geanakoplos [2012]. Haircuts

in these papers are pinned down by the price of Arrow securities, subject to additional constraints.

In my model, the repo contract is the only contract traded in the market, and the haircut is

determined by a necessary condition for equilibrium rollover. This complements the extensive liter-

ature on credit cycles, starting with Kiyotaki and Moore [1997a] and Bernanke and Gertler [1989],

with recent development including Adrian and Shin [2010], Brunnermeier and Sannikov [2012] and

Gertler and Kiyotaki [2013]. In all of these papers on credit cycles, the borrowing constraint is

exogenous and always binding.

Another feature of the repo market is that investors can build leverage through rehypothecation

(see Singh and Aitken [2010] for empirical evidence; see Bottazzi et al. [2012] for a theory of rehy-

pothecation). Additional issues introduced to the repo market by rehypothecation, such as novation

(see Du�e [2010]), is left for further research. Unsecured lending in a long-term relationship, as in

Kehoe and Levine [1993], is not considered here.

Disruption in the repo market between money market mutual funds (MMMF) and broker-

dealers also played a major role in the crisis. I abstract from these issues and focus instead on

the repo market between dealer banks. Gorton and Metrick [2012b] show that during the financial

crisis, MMMF did not reduce net lending to the repo market as a whole. The disruption in the

repo market took place in the bilateral repo market between broker-dealers or hedge funds and

broker-dealers. Unsecured lending though such markets as the Fed funds market is not allowed (see

Afonso and Lagos [2012] for a model about the market). Operational risk such as settlement fails

in the repo market (see Fleming and Garbade [2005]) is also not modeled.

The rest of the paper is organized as follows. Section 2 introduces in more detail institutional

features that the model aims to capture and stylized facts of the crisis that motivate the exercise.

Section 3 introduces the model. In section 4, I formally define the model’s dynamic equilibrium.

Section 5 discusses e�ciency of the model. In section 6, I characterize the equilibrium with rollover

of the repo contract, study the e�ciency gain of using repo contracts and characterize the e�ciency

6

loss from maturity mismatch and collateral risk. Section 7 studies the stability of the shadow

banking system by looking at the transition dynamics triggered by a small but unexpected increase

in collateral risk. Section 8 concludes the paper.

2 Shadow banking and the repo market: institutional features

and stylized facts

In this section, I explain in more detail institutional features and stylized facts of collateral risk,

the repo market, and the shadow banking system.7

Shadow banking and repo lending between financial intermediaries In many financial

markets, either the demand side or the supply side involves mainly agents from the real sector,

whereas in the repo market, many participants are financial intermediaries who could be on either

side of the market. In particular, broker-dealers lend to each other in the repo market as part of the

process of credit creation in the shadow banking system. Hedge funds and broker-dealers implement

arbitrage strategies with each other using securities lending contracts. This trade reduces the cost

of investment and arbitrage for the participants.

Credit creation in the shadow banking system typically starts with loan origination and loan

warehousing and ends with wholesale funding provided by such institutions as MMMF. Before the

loans reach the final cash suppliers, they need to be packaged into asset-backed securities (ABS),

collateralized debt obligations (CDO) and asset-backed commercial paper (ABCP), which typically

involves securities issuance, warehousing, tranching and intermediation.8 The intermediate stages

take time and rely on financing through the interbank lending market. An intermediation chain

comprised of broker-dealers and other intermediaries of the shadow banking system is formed during

the process. This intermediation chain within the shadow banking system relies heavily on short-

term collateralized loans, such as repo. Repo financing in these intermediate steps is not directly

financed by the final cash suppliers, but by other broker-dealers and financial intermediaries. This

market-based intermediation is what distinguishes a shadow banking system from a traditional7See, for example, Gorton and Metrick [2012a] and Copeland et al. [2012] for more details on the institutional

features related to the repo market and the financial crisis.8See Pozsar, Adrian, Ashcraft, and Boesky [2013].

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(commercial) banking system.

Maturity mismatch, solvency and liquidity of financial intermediaries Agents in the repo

and securities-lending markets rely on short-term debt to finance investments of longer maturity.9

The maturity mismatch links the liquidity of the short-term lending market to borrowers’ solvency.

In the case of Lehman Brothers, the crisis started from the asset side rather than from short-term

financing per se. The CDOs, – illiquid, long-term investments – that Lehman was initiating started

losing money and became hard to sell long before the crisis. To wait for the investment to turn

around, the bank had to roll over the debt. In the end, Lehman lost the race, ran out of collateral

and was forced into bankruptcy. The bankruptcy was not necessarily a self-fulfilling run. According

to a Wall Street Journal report: “Six weeks before it went bankrupt, Lehman Brothers Holdings

Inc. was e�ectively out of securities that could be used as collateral to back the short-term loans

it needed to survive.” And Lehman had to rely on “Repo 105”, a way to borrow against collateral

without exposing its high leverage to the public, as early as the end of 2007. Even without a run,

the bank may have had to default as it ran out of collateral.10

Speculation and arbitrage through securities lending also typically involve maturity mismatch.

Convergence trades, which involve going long on one asset and short on a similar asset, typically

involve maturity transformation, as a di�erence in liquidity is often associated with the spread

between similar assets, and it takes time to realize the arbitrage profit. If it takes longer than

expected to realize those gains from trade, the arbitrageurs may run out of funding and end up

insolvent, as Lehman Brothers did. The downfall of Long-Term Capital Management11 and MF

Global12 are similar cases.

In all these cases, the solvency of a financial intermediary depends on the maturity of investment

projects and the liquidity of the repo or securities lending market. The solvency and liquidity of a

financial intermediary is best understood in terms of an equilibrium model.9Adrian et al. [2009] shows that about 65% of outstanding repos of primary dealers are overnight repos.

10“Repos Played a Key Role in Lehman’s Demise”,

http://online.wsj.com/article/SB10001424052748703447104575118150651790066.html11http://en.wikipedia.org/wiki/Long-Term_Capital_Management#Downturn12http://en.wikipedia.org/wiki/MF_Global

8

Figure 1: Distribution of home price changes by county (from NYFed website:

http://www.newyorkfed.org/home-price-index/)

Collateral risk in the 2007-2008 financial crisis

According to the home-price-index of the New York Fed,13 the growth rate of housing prices slowed

down before 2007 and turned negative close to the end of 2007. Figure 1 illustrates the year-over-

year changes in housing prices in the US at the county level. In August 2007, 50% of counties

experienced negative price changes and at the end of 2007, more than 75% of counties started to

show a decline in housing prices. The grey area in the figure marks the great recession.

As housing prices spiraled downward, the riskiness of mortgage-backed securities (MBS) in-

creased. Figure 2 14 shows the market price index for a credit default swap contract that provided

insurance against the default risk of a pool of mortgage-backed securities issued in early 2006. The

discrepancy between the par value, 100, and the actual price index measures the shift in market be-

liefs about the riskiness of the mortgage-backed securities. The figure shows the index for tranches

with a AAA rating to tranches with a BBB- rating. Before mid-2007, the market’s belief about the13http://www.newyorkfed.org/home-price-index/14According to http://www.nera.com/nera-files/PUB_Subprime_Series_Part_IX_0412.pdf.

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Figure 2: ABX index and risk of mortgage backed securities.

riskiness of all tranches barely moved. Riskiness of tranches with lower ratings increased first, at

the beginning of 2007, and then in July 2007, riskiness of AAA tranches increased from zero to a

positive number and kept increasing. The sudden changes in the price index imply that the shift

in market beliefs came as an unexpected shock.

Why was there a sudden shift in the market belief about collateral risk? Dang et al. [2009,

2012] relate this to the information sensitivity of the debt contract and the lemons problem of

MBS. However, the link between the changes in housing prices and the riskiness of mortgage-

backed securities indicates that the risk is more likely to be related to the unexpected collapse

of the housing market in the whole country, rather than to the lemons problem that arises from

the quality deterioration of a fraction of mortgages. And the collapse of the housing market is so

widely publicized that asymmetric information on this fact was unlikely. Thus in this paper, I take

the view that the crisis was triggered by an unexpected increase in collateral risk, rather than by

market failure due to a lemons problem.

Balance sheet adjustments of financial intermediaries and liquidity of the bilateral

repo market He et al. [2010] estimates that, on the asset side, hedge funds and broker-dealers

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reduced holdings of securitized assets by approximately $800 billion during the 2008 crisis. It was

not just the size of broker-dealers’ balance sheets that changed; the composition changed as well.

Before the crisis, in November 2007, credit and mortgage-related assets made up of 32% of the

total value of the trading assets of Goldman Sachs, Morgan Stanley and Merrill Lynch. After the

crisis, in March 2009, they accounted for 23%.15 The flight to such safe assets as treasury bills

may be related to the increasing market risk of securities and is also consistent with their reduced

activity in the repo market, where the haircut for risky collateral assets increased sharply.16 Gorton

and Metrick [2012b] finds through Flow of Funds data that both the Repo assets and liability of

broker-dealers shrank during the crisis, indicating the freeze of the repo market and broker-dealers’

reduced activity in the market, as illustrated in Figure 3. This evidence implies that not only

financial intermediaries’ balance sheet, but also the portfolio composition of the balance sheet in

the shadow banking system, may have contributed to and been a�ected by the financial crisis.

Krishnamurthy et al. [2012] shows that funding from cash providers such as MMMF did not

change dramatically during the crisis. This implies that the dramatic change happened in the

bilateral repo market between broker-dealers and hedge funds, which is where increasing haircut

is reported in Gorton and Metrick [2012a,b] and Hördahl and King [2008]. This is consistent

with Gorton and Metrick [2012b]’s finding that repo lending of investors other than money market

mutual funds shrunk dramatically during the crisis. (See Figure 3 for dynamics in repo lending

and borrowing. See Figure 4 for dynamics in haircut.)

3 The model

The model is set in continuous time. The economy starts at t = 0 and lasts forever. There is a

continuum of agents of constant measure. At any moment, there is a constant inflow of entrants,

÷, and an equal outflow of exits.

There is a durable consumption good in the economy and there are some productive trees. A15See Table 7 of He et al. [2010]. The definition of a trading asset is “a collection of securities held by a firm

that are held for the purpose of reselling for a profit. Trading assets are recorded as a separate account from the

investment portfolio.” (http://www.investopedia.com/terms/t/trading-assets.asp)16For example, Gorton and Metrick [2012a] documents the devaluation of BBB asset-backed securities and a sharp

increase in haircut in the bilateral repo market during the crisis.

11

Broker-dealers’ repo assets and liabilities, in $billions

Figure 3: The freeze of the repo market. (from Gorton and Metrick [2012b])

Figure 4: Collateral quality and haircut dynamics during the crisis. (from Gorton and Metrick

[2009])

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tree bears consumption goods only at its maturity date. The maturity date of a tree is random

and follows an idiosyncratic Poisson process.

Agents are ex ante homogeneous. They are endowed with a0 œ R++ units of collateral trees

and m0 œ R++ units of consumption good when they enter the economy.

An agent’s expected payo� at time t is Et´ T

t cue≠fl(u≠t)du, where cudu is the measure of apples

she consumes between u and u + du, fl is the discount factor and T is the random moment when

she leaves the economy.

A collateral tree matures with Poisson rate µ œ R++. If a collateral tree matures at date t,

it bears y · Êt apples at maturity date t, with y œ R++. Êt is the aggregate state of the economy

at date t; it represents the aggregate risk that the quality of a collateral asset may deteriorate, or

the aggregate collateral risk. There are two aggregate states, Êt œ {0, 1}. When Êt = 1, every tree

bears y apples if it matures at t, and when Êt = 0, every tree bears no apples. I assume that the

economy is in the good state initially, Ê0 = 1. And the bad state, Ê = 0, is assumed to be an

absorbing state. The arrival of the bad state follows a Poisson process with rate ‰ œ R++. As the

likelihood of shock, Êt = 0, can be small or large, the model applies to collateral assets of both

high and low quality.

Another type of tree in the economy represents investment opportunities. An investment op-

portunity is a long-term technology that transforms consumption goods at the investment date into

consumption goods at the (random) maturity date. The maturity date arrives with Poisson rate

fi œ R++. The investment is one-shot and doesn’t require additional resources at subsequent dates.

If an agent does not exploit the investment opportunity the moment it arrives, she loses it. With

i units of consumption good as input at the investment date, the output at maturity is fl+fifi f(i),

where f : R++ æ R++. In the benchmark model, I assume that the production function takes the

form, f(i) = ◊i–, with productivity parameter ◊ œ R++ and – œ (0, 1), so f(i) is concave and the

marginal output at zero input is infinity. I assume that only the agent who invests in a long-term

technology has the skill to manage it. The output from the technology if other agents own the

project is zero, and other agents cannot take the output from it away from the investor.

The long-term investment opportunity is endowed to agents with delay, which represents the

search friction to find a profitable investment opportunity. After entering the economy, each agent

receives at most one investment opportunity, at a random date. The arrival date of an agent’s

13

Figure 5: The life cycle of an agent in Autarky.

investment opportunity follows an idiosyncratic Poisson process, arrival rate ⁄ œ R++.

An agent leaves the market after her collateral asset and the long-term project mature. I assume

for simplicity that agents lose their chance to find an investment opportunity after their collateral

matures.

Figure 5 describes a realization of the life cycle of an agent in Autarky. Upon entry, the agent

decides her consumption and storage. She stores some of the consumption good in order to provide

for investment in the long-term technology. When the investment opportunity arrives, she draws

her consumption good from storage to invest in the project and then waits for the project to mature.

She leaves the market after both her asset and the project mature.

Since storing consumption is not productive and delays consumption, a financial system can

improve e�ciency by allocating consumption goods in storage to agents who need more funding to

take advantage of the investment opportunity.

The repo market allows agents waiting for long-term projects to earn interest by lending, and

agents with long-term projects can increase their investment by borrowing against their collateral.

The repo market is assumed to be perfectly competitive. A repo contract has three components:

the interest rate Rt œ R+, haircut ht œ [0, Œ) and maturity dt. According to the contract, a

borrower puts down ht units of collateral for each unit of consumption she borrows from the lender

at the moment of signing the contract. At maturity, date t+dt, if no party defaults on the contract,

14

then the borrower pays 1 + Rtdt units of consumption good for each unit she borrows at t, and the

lender delivers ht units of collateral asset back to the borrower for each unit of borrowing.17 Given

the contract, an agent with asset holding a can borrow up to aht

.

At the maturity date of a repo contract, the borrower can choose to default or repay the debt.

There are two ways to repay the debt: by using consumption goods that the borrower stored from

her endowment or that she obtained when her tree matures; or by borrowing from other lenders.

This is what I refer to as repo rollover. As long as a borrower has not borrowed up to aht

, she can

choose to roll over her debt.

For lenders, as in Acharya and Bisin [2013], I assume that the repo market is an opaque, over-

the-counter market, so the repo contract is not conditional on information such as the borrowers’

balance sheet.18 At any moment, one repo contract clears the whole market. I assume that every

lender is assigned to one borrower in the market-clearing process.19 After the repo market clears,

lenders meet their counterparty. At that moment, lenders can see borrowers’ information and

whether or not they will default. Then, lenders can decide whether to carry out the repo contract

or to reject the borrower and wait until next period.

Since a lender is matched with one borrower, the counterparty risk, the risk that a borrower may

default, is undiversified. Denote the probability that the counterparty defaults on a repo contract

signed at t as ”tdt. Counterparty default is a shock to the lender’s portfolio. For a lender with a

units of asset and s units of apple in repo account, she will hold a + hts units of asset but will have

no funding left in the repo account when default happens. If default happens with probability one,

the contract is observationally equivalent to a spot transaction in which the borrower sells the asset

to the lender, at unit price 1ht

.

To simplify the analysis, I impose the following restrictions on agents’ strategies. A borrower

is not allowed to borrow or lend in the repo market after she defaults. A lender is not allowed

to move additional funding from storage to the repo market after she loses all her funding to the17Default does not incur any loss to the defaulting agent other than the collateral asset in the repo contract. For

example, for an agent with a long-term investment, default does not a�ect her return from the investment because

the investment generates only private return for the borrower that no one else can control.18Trading delay due to search friction, as inDu�e et al. [2005], Lagos and Rocheteau [2009], Afonso and Lagos

[2012], is also ignored here.19A lender can only lend to one borrower, a borrower can borrow from several lenders.

15

repo market when she meets a defaulting borrower or she herself defaults on a repo contract.20

With these two simplifications, we focus on the borrowing decision of agents who need to finance

their long-term investment and lending decisions of those lenders who have not met a defaulting

borrower. Additionally, I assume that interest payments from repo lending must be consumed and

cannot be accumulated for repo lending or storage.

3.1 Discussion

If all borrowers default with probability one, the market is equivalent to a market for trading the

asset; however, agents cannot choose to buy or sell the asset and lend or borrow against the asset.

What is missing is an additional market for a firesale asset. In such a case, lenders do not worry

about counterparty default risk, and buyers of the asset can optimize their portfolio based on their

risk exposure. While the current setup is in line with the observation that the repo market is much

more liquid than the market for trading assets, I will study the e�ect of introducing an additional

market in the extension.

I also assume that the default over the collateral does not a�ect agents’ payo� from their project

investment. Default does not lead to bankruptcy, and collateral delivery is the only requirement if

a borrower defaults. The separation of default from bankruptcy allows the agent to buy and sell

the collateral asset through repo contracts (and default). And this is consistent with the exemption

of repo contracts from “automatic stay” (Garbade [2006]). The separation also allows the model

to characterize credit derivative markets. I also assume that default does not incur loss to the

borrower other than the collateral she puts down because other investors cannot separate he from

her long-term investment. This is consistent with moral-hazard frictions that lenders may face.

Another property of the repo contract is that dividends from the collateral asset before the

maturity of the repo contract belong to the lender. This is not an issue for overnight repo, as no

dividend is generated from an asset overnight. Likewise, since the maturity of a repo contract is

assumed to be infinitesimal in my model, there will not be any dividend payment from the asset

before the maturity of the contract.

I assume that lenders face undiversified counterparty risk. Although banks in the real world20When losing her funding of amount s to the repo market, she has s(1 + h) additional collateral asset. But she

can still borrow from the repo market.

16

are large enough to diversify the idiosyncratic coutnerpary risk in normal times, diversification

would be impossible when default is triggered by certain aggregate shocks. My assumption will

be more useful when studying the dynamic response of the financial system to aggregate shocks.

Another reason I make this assumption is that I am modeling the repo market between financial

intermediaries. As financial intermediaries borrow and lend on a large scale and the total number of

financial intermediaries is limited, undiversified counterparty risk is more relevant for the inter-bank

repo market.

4 Equilibrium definition

In this section, I give the formal definition of the dynamic equilibrium with a certain initial distribu-

tion of agents and initial aggregate state. In the equilibrium definition subsection, I first formalize

the individual agent’s problem, after which I define the law of motion of the economy. And in the

end, I define the equilibrium.

4.1 An agent’s problem

An agent’s problem depends on whether or not the repo market functions, which, in turn, depends

on the aggregate state.

When Êt = 0, the repo market is not functioning, as the supply of valuable collateral asset in

this state is 0.

When Êt = 1, the repo market is functioning. Then, an agent’s problem depends on her portfo-

lio, her lending/borrowing history and whether she is still looking for an investment opportunity or

she has already invested in a project. In that case, the life cycle of an agent who manages to find a

long-term project is illustrated in Figure 6. After entry, the agent allocates her consumption good

endowment to consumption, storage and repo lending. Before she finds her long-term technology,

she continues making decisions on consumption, storage and repo lending.21 When she finds her

long-term project, she withdraws consumption goods from storage and repo lending, borrows from

the repo market and invests in the long-term project. Before her long-term investment matures,

she does not have consumption goods left so she decides whether to roll over her debt or default. If21When lending to other agents, the agent may meet a default borrower with Poisson rate ”. If that happens, her

lending turns into additional asset holdings and she loses her capacity to continue lending.

17

Figure 6: life cycle of an agent with access to the repo market

her long-term investment or asset matures when she is still rolling over her borrowing, she repays

her outstanding debt, consumes the remaining consumption and waits for the rest of her trees to

mature. After all her asset matures and she consumes all the consumption she owns, she leaves the

market.

To solve an agent’s problem when Êt = 1, I go backward. I first solve her problem after project

investment; then, I solve her problem at long-term investment, her problem before the investment,

and, finally, her problem at the beginning of her life. I then explain an investor’s problem when

Êt = 0. Table 1 summarizes the value function and policy functions for an agent’s problem when

she faces di�erent situations. I now explain an agent’s problem in di�erent situations.

Situation 1: Êt = 1, agent’s problem after LT investment

Wt(a, m, s, i) = maxc,mÕ,sÕ

c + EWt(a, mÕ, sÕ, i), (1)

s.t.c + mÕ + sÕ Æ s + m,

sÕ Ø ≠ aht

,

c, mÕ Ø 0,

where the first constraint is the agent’s budget constraint; the second is the collateral constraint,

18

situation value function policy function distribution

Êt = 1, after LT investment Wt(a, m, s, i)

consumption c1t

storage m1t

net lending s1t

acceptance z1t,

default d1t

F1t(a, m, s, i)

Êt = 1, at LT investment Ut(a, m, s)

consumption cIt

storage mIt

net lending sIt

investment iIt

0

Êt = 1, active before LT investment Vt(a, m, s)

consumption c0t

storage m0t

net lending s0t

acceptancez0t

default d0t

F0t(a, m, s)

Êt = 1, deactivated before LT investment V dt (a, m)

consumption cdt

storage mdt

Fdt(a, m)

Êt = 0, before LT investment V At (a, m)

consumption cA

storage mA

G0t(a, m)

Table 1: Value functions and policy functions

19

which imposes that borrowing cannot exceed as

1+h ; and the last two are non-negativity constraints

for consumption and storage allocation. Agents choose consumption c, storage mÕ and repo lending

sÕ to maximize the payo� from consumption and continuation value EWt(a, mÕ, sÕ, i), which depends

on random events between t and t + dt.

EWt(a, m, s, i) (2)

=µdt5a + m + s + f(i)

6+ fidt

5µ

fl + µ + ‰a + m + s + fl + fi

fif(i)

6+ ‰dt

5m + f(i)

6

+”tdtIs>0e≠fldt maxzœ[0,1]

5zWt+dt(a + hts, m, 0, i) + (1 ≠ z)Wt+dt(a, m + s, 0, i)

6

+ (1 ≠ (µ + fi + ‰ + ”tdtIs>0)dt) e≠fldt maxdœ[0,1]

5d

3µy

fl + µ + ‰(a + hts) + m + f(i)

4

+ (1 ≠ d) (Rtdts + Wt+dt(a, m, s, i)) Is>0 + (1 ≠ d)Wt+dt(a, m, (1 + Rtdt)s, i)IsÆ0

6.

The continuation value depends on several random events. With probability µdt, the agent’s asset

matures. In this case, she repays ≠s apples to her lenders, if s < 0, or withdraws s apples from

inter-bank lending, consumes a+m+s apples, and waits for her long-term project to mature, which

delivers expected payo� f(i). With probability fidt, the agent’s long-term investment matures. In

this case, she repays her debt or draws lending from the repo market, consumes m + s + fl+fifi f(i)

apples, and waits for her collateral asset holdings to mature, which delivers expected payo� µfl+µ+‰a.

With probability ‰dt, the aggregate devaluation shock hits the economy and the collateral asset

becomes worthless. So if s < 0, she default on her debt, consumes m apples that she withdraws

from her storage account and waits from her long-term investment to mature. If s > 0, on seeing

the aggregate shock hitting the economy, she withdraws her apples from the repo market and

consumes the s apples immediately. The next term is the expected payo� from the event that the

counterparty defaults. This term shows up only if the agent is lending in the repo market. So, if

s > 0, then with probability ”tdt, the agent meets a defaulting borrower. In this case, her payo�

depends on whether she is willing to lend to the borrower after knowing that the borrowing is going

to default. If she decides to lend to the defaulting borrower, her loan s will turn into hts units of

additional asset holdings when the repo contract matures in dt period. Thus, her continuation value

with this choice is e≠fldtW (a + hts, m, 0, i). If she decides not to lend to the defaulting borrower,

she adds s additional apples to storage account for dt period. So her continuation value with this

choice is e≠fldtW (a, m + s, 0, i). With the residual probability, [1 ≠ (µ + fi + ‰ + ”tdtIs>0)dt], the

20

agent meets a non-defaulting borrower and decides whether or not to default. If she defaults, her

continuation value is e≠fldtË

µy(a+hts)fl+µ+‰ + m + f(i)

È, where the first component is the expected payo�

from the asset holdings, m is the payo� from the consumption of apples withdrawn from the storage

account and f(i) is the expected payo� from the long-term investment. If she does not default, she

will get Rtdts apples as interest payment, so her continuation value is e≠fldtW (a, m, (1 + Rtdt)s, i).

Refraining from lending is ruled out here as long as interest rate Rt is positive. If s < 0, she needs

to borrow from the repo market. Similarly, e≠fldtÓ

µy[a+hts]fl+µ+‰ + m + f(i)

Ôis her payo� from default,

and e≠fldtW (a, m, (1 + Rtdt)s, i) is her payo� from rolling over her debt.

Debt rollover is an option only if (1 + Rtdt)s Ø ≠ aht

. So, if the state variable s = ≠ aht

, the

agent’s borrowing constraint is already binding, so she must default. Thus,

EWt

3a, m, ≠ a

ht, i

4= m + f(i), (3)

which is equal to m + f(i) plus some term of the same order of magnitude as the infinitesimal

period dt.

Situation 2: Êt = 1, agent’s problem at long-term investment

Ut(a, m, s) = maxc,mÕ,sÕ,i

c + Wt(a, mÕ, sÕ, i) (4)

s.t.c + sÕ + mÕ + i Æ s + m,

sÕ Ø ≠ aht

,

c, mÕ, i Ø 0,

where agents choose consumption, c, storage, mÕ, lending or borrowing through the repo market,

sÕ, and investment in the long-term project, i, to maximize their payo� from consumption and con-

tinuation value Wt(a, mÕ, sÕ, i). The constraints the agent faces are resource constraint, borrowing

constraint and non-negativity constraints of the choice variables, c, mÕ and i.

21

Situation 3: Êt = 1, agent’s problem before long-term investment

Vt(a, m, s) = maxc,sÕ,mÕ,z,d

c + EVt(a, mÕ, sÕ), (5)

s.t.c + sÕ + mÕ Æ s + m,

sÕ Ø ≠ aht

,

c, mÕ Ø 0,

which is similar to agents’ problem after project investment, equation 1.

EVt(a, m, s) (6)

=µdt (a + m + s) + ⁄dtUt(a, m, s) + ‰dtV At (m)

+”tIs>0dte≠fldt maxzœ[0,1]

5zV d

t+dt(a + hts, m) + (1 ≠ z)Vt+dt(a, m, s)6

+ [1 ≠ (µ + ⁄ + ‰ + ”tIs>0)dt] maxdœ[0,1]

e≠fldt5dVt+dt(a + hts, m, 0)

+ (1 ≠ d) (Rtdts + Vt+dt(a, m, s)) Is>0 + (1 ≠ d)Vt+dt(a, m, (1 + Rtdt)s)IsÆ0

6.

The continuation value has a similar expression to the continuation value 2. The additional random

event is the event of finding a long-term project. With probability ⁄dt, the agent finds a long-

term project in dt period. The continuation value contingent on the event is Ut(a, m, s). With

probability µdt, the agent’s asset matures.22 And if a lender lends to a defaulting borrower, she

will be deactivated, with continuation value, V dt+dt(a + hts, m).

If s = ≠ aht

, the agent’s borrowing constraint is already binding, then she must default. So,

EVt

3a, m, ≠ a

ht

4= V A

t (m). (7)

After default, the agent has no asset left. So her continuation value is the same as what she would

have been under Autarky with storage m.

The problem of an agent deactivated because of lending to a defaulting borrower is22The collateral asset of an agent is assumed to mature at the same time. This is not exactly consistent with the

assumption that the maturity of collateral asset is idiosyncratic across agents because the asset an agent receives

from other agents is assumed to mature at the same time as his own asset.

22

V dt (a, m) = max

c,mÕc + EV d

t (a, mÕ), (8)

s.t.c + mÕ Æ m,

c, mÕ Ø 0,

where the continuation value is,

EV dt (a, m) = µdt (a + m) + ⁄dtUt(a, m, 0) + ‰dtV A

t (m) + [1 ≠ (µ + ⁄ + ‰)dt] e≠fldtV dt (a, m). (9)

The agent finds her long-term project with probability ⁄dt and her continuation value is Ut(a, m, 0).

Situation 4: Êt = 0

The value function, VA(m) and policy functions, cA(m), mA(m), solve the problem of an agent’s

problem before she finds her long-term projects.

VA(m) = maxc,mÕ

c + EVA(mÕ), (10)

s.t.c + mÕ Æ m,

c, mÕ Ø 0,

where the agents choose consumption c and storage for the next period mÕ to maximize their payo�

from consumption and continuation value EVA(mÕ), subject to the resource constraint and non-

negativity constraints on consumption and storage. The continuation value depends on the random

events that may happen during dt period.

EVA(m) = ⁄dt max0ÆiÆm

{m ≠ i + f(i)} + µdtm + [1 ≠ (⁄ + µ)dt] e≠fldtV A(m).

With probability ⁄dt, the agent finds her long-term investment and, in that case, chooses optimally

to invest in the long-term project and consumption so as to maximize her expected payo� c + f(i),

where c = m ≠ i, subject to non-negativity constraints of consumption and investment. With

probability µdt, her collateral asset matures and bears no apples at maturity and she loses her

chance to find a long-term investment opportunity in the future. In that case, she consumes away

the apples in the storage account and leaves the market. With the residual probability 1≠(⁄+µ)dt,

nothing happens during the dt period so her continuation value is e≠fldtV A(m).

23

Figure 7: Law of motion of an agent’s state before the realization of collateral shock (Êt = 1).

Laws of motion for the distribution of agents

The exact law of motion is left to the Appendix. Here, I illustrate the laws of motion on the

equilibrium path, where Êt = 1, ’t using Figure 7. All new entrants enter as active lenders and

are counted in the distribution F0t. A lender leaves the pool of active lenders in three situations.

If her collateral asset matures before she finds her long-term project, she leaves the economy after

consuming all her consumption goods. If she meets a defaulting borrower, she enters the pool

of deactivated lenders with distribution Fdt. If she finds her investment opportunity, she enters

the pool of borrowers with distribution F1t. An agent in the pool of deactivated lenders leaves the

economy if her collateral asset matures before she finds a project, or she enters the pool of borrowers

when she finds her long-term project. An agent in the pool of borrowers exits the economy after

both her project and collateral asset mature. She will stop borrowing when her collateral asset or

project matures, or when she defaults on her loan.

Equilibrium definition

Definition 4.1. An equilibrium with initial distribution F10, Fd0 and F00 and initial state Ê0 = 1,

is the repo contract term {Rt, ht}tØ0, default rate, {”t}tØ0, agents’ policy functions and value

functions and aggregate law of motion such that,

24

(i) given R, h and ⁄, agents’ policy functions and value function solve their problems if Êt = 1.

(ii) agents’ policy functions and value functions solve their problem if Êt = 0.

(iii) the contract (R, h) clears the repo market given the distribution of agents and agents’

decision function,

ˆs1t(a, m, s, i)dF1t (a, m, s, i) +

ˆs0t(a, m, s)dF0t (a, m, s) = 0, if Êt = 1, ’t.

(iv) the distribution of agents is endogenously determined by laws of motion.

(v) agents’ expectation on default rate ”t is consistent with the actual default rate,

”tdt =´

{(a,m,s):sÆ0} d1t(a, m, s, i)sdF 11t (a, m, s, i) +

´{(a,m,s):sÆ0} d0t(a, m, s)sdF 1

0t (a, m, s)´{(a,m,s):sÆ0} sdF 1

1t (a, m, s, i) +´

{(a,m,s):sÆ0} sdF 10t (a, m, s) , for Êt = 1.

”tdt is the probability that a lender’s counterparty may default on her borrowing. Since the

lender is ignorant of the borrower’s portfolio, the lender’s counterparty can be thought of as a

random draw from the pool of borrowers, weighted by borrowers’ funding demand. Suppose that

an agent borrows to invest in a long-term project, and she also has invested all her apples in

storage and in interbank lending. When the initial borrowing matures in dt period, the cumulative

probability that her project or collateral asset matures is (fi +µ)dt, which is negligible. She has two

choices, to roll over her debt or to default. If every borrower chooses to default immediately after

initial borrowing, ”tdt = 1. ”tdt < 1 when some agents choose to rollover their debt in equilibrium.

Definition 4.2. A stationary equilibrium with initial state Ê0 = 1 is an equilibrium with initial

distributions F10, Fd0 and F00 and initial state Ê0 = 1, such that F0t (a, m, s) = F00 (a, m, s),

Fdt(a, m) = Fd0(a, m), and F1t (a, m, s, i) = F10 (a, m, s, i), ’t.

5 E�ciency

To understand the welfare loss from the market incompleteness, I study properties of e�cient

allocation in this section.

The social planner’s choice is aggregate consumption flow, C· , for each newly born agent at

moment · , aggregate storage, m· , and investment in long-term tech, i· , for each agent with an

opportunity to invest in the long-term technology.

The social planner’s problem at moment t is

25

St = max{C· ,m· ,i· }’·Øt

ˆ Œ

tC· e≠fl(·≠t)d·,

s.t.C· d· + (⁄d·) ÷

µ+⁄ i· + m· Æ´ ·

≠Œfl+fi

fi ◊i–s (fid·) e≠fi(·≠s)⁄ ÷

µ+⁄ds

+´ ·

≠Œ ayÊ· (µd·) e≠µ(·≠s)÷ds + (÷d·) M· + m·≠d· ,

C· , m· , i· Ø 0.

In the objective function of the planner, ÷dt is the measure of agents born during · and · +

dt, and e≠fl(·≠t) is the discount factor for future welfare gain. The planner chooses allocation,

{C· , m· , i· }’·Øt, to maximize the objective function, subject to nonnegative constraints of the

choice variables and the resource constraint. The right-hand side of the resource constraint includes

apples from maturing long-term projects from past investments, maturing collateral from agents

born before · , and production from short-term technology and storage technology. fl+fifi ◊i–

s is the

measure of fruits from a maturing project created at moment s with investment is. (fid·) e≠fi(·≠s)

is the probability that a project created at moment s matures between · and · + d· . And ⁄ ÷µ+⁄ds

is the measure of projects created between moment s and s + ds. Similarly, ayÊ· is the measure

of fruits from the maturing asset of an agent. (µd·) e≠µ(·≠s) is the probability that the asset of an

agent born at moment s matures between · and · + d· . And ÷ds is the measure of agents born

between s and s+ds. The left-hand side of the resource constraint is the allocation of the available

apples to consumption, long-term investment and storage. (⁄d·) ÷µ+⁄ is the measure of projects

found between · and · + d· .

Proposition 5.1. The e�cient allocation {C· , m· , i· }’·Øt is

i· = iú, such that ◊–iú–≠1 = 1,

m· = 0.

And the e�ciency of the economy is characterized solely by investment allocation iú

C· =5

fl + fi

fi◊iú– ≠ iú

6⁄÷

µ + ⁄+ ayÊ· ÷.

As a benchmark for comparison, in Autarkic allocation, where the repo market is shut down,

individual investment, i, is constrained to be equal to the amount of apples agents are endowed with

and store using the storage technology until they find the project. If the initial endowment is small

26

enough so that the marginal return from storing all endowments is greater than 1, ⁄◊–m–≠10 +µ

fl+µ+⁄ > 1,

the expected return from investment is ⁄◊m–0 +µm0

fl+µ+⁄ . Otherwise, the storage is such that the expected

marginal return from storage is equal to 1, ⁄◊–m–≠10 +µ

fl+µ+⁄ = 1. So, the marginal return from investment

in long-term technology is greater than fl+⁄⁄ . Therefore, there exists a wedge between the marginal

return of the project and the marginal utility of consumption in Autarkic allocation, which is

greater than fl+⁄⁄ ≠ 1 = fl

⁄ . The lower bound of the wedge increases with the search friction to find

the long-term project.

In the e�cient allocation, in contrast, the wedge between the marginal return of the project

investment and the marginal utility of consumption is 0. The social planner, not subject to en-

forceability constraints, can allocate consumption goods from maturity projects to new investment.

Thus, the optimal investment allocation does not depend on the aggregate risk on the value of

collateral. The e�cient investment in the long-term project tree does not depend on the value of

the collateral asset. As we will see, the e�ciency gain from using repo contracts in this environment

comes from the transfer of output from maturity projects to investment in new projects.

Another di�erence between Autarkic allocation and the first-best allocation is the allocation to

storage, which can be interpreted as liquidity hoarding. While the storage of each agent in Autarky

is equal to her investment in long-term technology, the aggregate storage is 0 in e�cient allocation.

In other words, it is socially wasteful for agents to hoard liquidity when the social planner is not

constrained by market incompleteness. The wedge and ine�ciency in the Autarkic economy also

comes from the fact that the return on storage technology is low, so it is costly for agents to use

the storage technology. If the return on storage technology is equal to the discount factor, the

lower bound of the wedge in Autarky would be zero. This is the case, for example, if the storage

technology is a fiat currency and the monetary authority follows Friedman rule. 23

6 Equilibrium characterization

With collateral risk and the state-contingent repo contract, default does not happen in equilibrium

as long as not all agents default. Next, I add maturity mismatch to the analysis. In the full

equilibrium, I will show that maturity mismatch results in repo rollover and equilibrium default.23See Lagos and Wright [2005], Berentsen et al. [2007] and Williamson and Wright [2010].

27

I will also show that default triggers more default in equilibrium because of the undiversified

counterparty risk.

6.1 Characterization of an equilibrium with repo rollover

With a short-term repo contract not contingent on liquidity arrival, agents borrow from the repo

market when they find their long-term investment opportunity, and then they roll over their debt

to wait for their long-term investment or collateral asset to mature. But as long as the interest rate

R is positive, they cannot roll over their debt infinitely. At certain point, they will reach their debt

limit, ah . The dynamics of debt holding in the equilibrium with debt rollover is illustrated in Figure

8(a). Liquidity arrives when the borrower’s collateral asset or project matures, which is a random

date that could be earlier or later than the moment of reaching the debt limit. The borrower will

keep rolling over her debt until she repays is using consumption goods from her trees at maturity

or she defaults when she reaches her debt limit.

In this equilibrium, there is a trade-o� between default and investment in long-term technology.

This trade-o� is illustrated in Figure 8(b). As initial borrowing bÕ increases, the moment of reaching

the debt limit moves to the left of the timeline. Thus, the probability that the borrower reaches

her debt limit before she receives liquidity from maturing trees decreases. If the agent takes on

more initial debt, she gains more from long-term investment but is less likely to receive apples from

long-term investment or collateral asset before she defaults when she reaches her debt limit.

(a) debt rollover (b) tradeo� for initial borrowing

Figure 8: debt rollover

28

The dynamics of other choice variables in the equilibrium with stationary distribution are as

follows. Before an agent finds her long-term investment, she keeps lending a constant amount

of apples to other agents through the repo market and stores a constant amount of apples. She

consumes all interest payments from lending. If she finds a long-term technology, she invests all

her apples in storage and repo lending and from her initial borrowing to the long-term technology

and stops consuming until the project or her assets mature. If she meets a defaulting agent before

her investment, all her lending turns into assets and her consumption drops to zero.

Necessary conditions for an equilibrium with debt rollover

Lemma 6.1. A necessary and su�cient condition for a borrower with debt holding b to roll over

her debt until reaching her debt limit is µfl+µ+‰yh Ø µ+fi

fl+‰+µ+fi≠R and R < fl + ‰ + µ + fi.

Through debt rollover, a borrower can avoid losing collateral assets to lenders. The benefit

of avoiding default is higher when haircut is high enough. Lemma 6.1 gives the necessary and

su�cient condition for borrowers to roll over their debt holdings.

For equilibrium rollover to take place, a lender must be willing to lend to defaulting borrowers,

and she does not default on repo lending to non-defaulting borrowers. This means that when she

meets a defaulting borrower, her continuation value from lending to that borrower, e≠fldtV d(a +

hs, m), must be no less than her continuation value from waiting for the next lending opportunity,

e≠fldtV (a, m, s). And when she meets a non-defaulting borrower, her continuation value from

waiting for the debt repayment, e≠fldtV (a, m, s(1 + Rdt)), must be no less than her continuation

value from defaulting e≠fldtV d(a + hs, m).

Lemma 6.2. A su�cient and necessary condition for lenders’ strategy in the equilibrium with debt

rollover to be optimal is: V (a, m, s) = V d(a + hs, m).

Lemma 6.2 implies that

µ (yh ≠ 1) s = sR + ⁄ [U(a, m, s) ≠ U(a + hs, m, 0)] ,

which shows that haircut compensates for two losses from lending to a defaulting borrower instead

of a non-defaulting borrower: the loss of interest payment; and the di�erence in the continuation

29

value when she invests in long-term technology. When the arrival rate of the aggregate shock

increases, the heterogeneity increases and haircut may be more likely to satisfy the condition for

borrowers to roll over their debt in Lemma 6.1.

Equilibrium characterization

Lemma 6.3. If the repo contract satisfies the condition in Lemma 6.1, the value function W (a, 0, ≠b, i)

for b œ R++in the equilibrium with debt rollover is characterized by the following di�erential equa-

tion

flW (a, 0, ≠b, i) = fi5

fl + fi

fif(i) + µ

fl + µ + ‰ya ≠ b ≠ W (a, 0, ≠b, i)

6(11)

+ ‰5f(i) ≠ W (a, 0, ≠b, i)

6+ µ

5f(i) + ya ≠ b ≠ W (a, 0, ≠b, i)

6

+ ˆW (a, 0, ≠b, i)ˆb

bR

and boundary condition

W3

a, 0, ≠a

h, i

4= f(i). (12)

The solution to the di�erential equation is

W (a, 0, ≠b, i) =5

µ + fi

fl + ‰ + µ + fi ≠ R

1h

≠ µ

fl + µ + ‰y

6a

3hb

a

4(fl+‰+µ+fi)/R

. (13)

≠ µ + fi

fl + ‰ + µ + fi ≠ Rb + f(i) + µ

fl + µ + ‰ya

The initial borrowing at the moment the agent receives an investment opportunity is pinned

down by problem (4). The trade-o� between losing assets through default and more investment in

the long-term project leads to the following result about initial borrowing.

Lemma 6.4. If h > µ+fifl+‰+µ+fi≠R

fl+µ+‰µy , f Õ(s+m) > µ+fi

fl+‰+µ+fi≠R and f Õ(s+m+ ah) < µ

Rfl+‰+µ+fi

fl+µ+‰ yh≠µ+fi

R , an investor’s initial borrowing, b, is solved by

fl + ‰ + µ + fi

R

3hb

a

4(fl+‰+µ+fi≠R)/R

=f Õ(s + m + b) ≠ µ+fi

fl+‰+µ+fi≠Rµ

fl+µ+‰yh ≠ µ+fifl+‰+µ+fi≠R

. (14)

Initial borrowing b = 0 if f(s+m) Æ µ+fifl+‰+µ+fi≠R and b = a

1+h if f Õ(s+m+ ah) Ø µ

Rfl+‰+µ+fi

fl+µ+‰ yh≠ µ+fiR .

30

The first condition for Lemma 6.4 is the su�cient and necessary condition for debt rollover

stated in Lemma 6.1. If the marginal return from long-term investment when an agent takes

on no initial borrowing is lower than µ+fifl+‰+µ+fi≠R , she will find it not profitable to take on any

initial borrowing. If the marginal return from long-term investment when she takes on her initial

borrowing up to the debt limit is higher than µR

fl+‰+µ+fifl+µ+‰ yh ≠ µ+fi

R , she will borrow up to her debt

limit.

Given the initial borrowing, the duration between an agent’s initial borrowing and reaching the

debt limit, T (a, b), depends on her initial borrowing and asset holding. It is solved by equationah = beRT (a,b). So, T (a, b) = 1

R ln! a

hb

". Given T (a, b), the probability that an agent is able to pay

back the debt isˆ T (a,b)

0(µ + fi)e≠(‰+µ+fi)tdt = µ + fi

‰ + µ + fi

Ë1 ≠ e≠(‰+µ+fi)T (a,b)

È= µ + fi

‰ + µ + fi

S

U1 ≠3

hb

a

4 ‰+µ+fiR

T

V .

Therefore, the probability of default is ‰‰+µ+fi + µ+fi

‰+µ+fi

1hba

2 ‰+µ+fiR , which is increasing in the ratio

between initial borrowing and the debt limit, b = hba . From equation (14), b is increasing in the

marginal return from project investment, f Õ(s + m + b), and decreasing in haircut and dividends

from an asset. This reflects the tradeo� between the return from project investment and losing

assets through default. So the default probability of an agent is increasing in the productivity of

the long-term technology and decreasing in haircut and the dividends of an asset.

The default probability of an agent after initial borrowing also depends on the liquidity of an

agent’s portfolio. For an agent with portfolio (a + hsÕ, m, s ≠ sÕ), the following corollary shows that

the default probability is increasing in sÕ.

Corollary 6.1. Default triggers more default. Counter-party default increases default probability

of lenders in the future. For an agent with portfolio (a + hsÕ, m, s ≠ sÕ),

db

dsÕ > 0

Corollary 6.1 is derived from Lemma 6.4. The portfolio of an agent with portfolio (a, m, s)

turns into (a + hs, m, 0) after she meets a defaulting agent. So, according to the corollary, an agent

whose counterparty defaults before she starts her long-term investment is more likely to default

than an agent who has not met a default counterparty when she starts her long-term investment.

31

Figure 9: intertemporal chain of borrowing and lending

This implies that counterparty default has an externality on the other agent’s default probability.

Figure 9 illustrates this externality. Agent i borrows from agent j and delivers her asset holding

to agent j upon default. This increases agent j’s default probability when she borrows from agent

k, among other agents, to invest in her long-term project. So, agent j is more likely to deliver

her assets to agent k upon default. As we can see from the inter-temporal chain of reactions, the

default of one agent increases the default probability of those agents who lend to her, which may

a�ect the default probability of those who lend to agent j’s. This intertemporal chain is reminiscent

of Kiyotaki and Moore [1997b], which studies the propagation of shocks through credit chains. In

my model, shocks to agents’ portfolio – making their portfolio less liquid – are contagious. The

increase in the probability of default passes on from a defaulting borrower to her lenders or, so to

speak, borrowers-to-be. The risk-taking of an individual borrower, therefore, adds to the risk of

the whole system. I will explore this type of contagion more in the section on dynamics.

Proposition 6.1. A stationary equilibrium with debt rollover must satisfy conditions in Lemma

6.1 and Lemma 6.2. The equilibrium can be summarized by interest rate R, haircut h, default rate

32

”, the portfolio choice of active lenders, (m, s), the initial borrowing of active lenders, b1 and the

initial borrowing of deactivated lenders, b0, in the following system of equations,3

hb1a0

4(fl+‰+µ+fi≠R)/R

= R

fl + ‰ + µ + fi

f Õ(m0 + b1) ≠ µ+fifl+‰+µ+fi≠R

µfl+µ+‰yh ≠ µ+fi

fl+‰+µ+fi≠R

, (15)

3hb0

a0 + hs

4(fl+‰+µ+fi≠R)/R

= R

fl + ‰ + µ + fi

f Õ(m + b0) ≠ µ+fifl+‰+µ+fi≠R

µfl+µ+‰yh ≠ µ+fi

fl+‰+µ+fi≠R

, (16)

µ (yh ≠ 1) s = sR + ⁄ [U(a, m, s) ≠ U(a + hs, m, 0)] , (17)

” = Bd

B, (18)

1µ + ⁄ + ”

s = B, (19)

(m, s) œ arg maxm+sÆm0,mØ0,sØ0

V (a, m, s). (20)

where B denotes total borrowing, Bd denotes the borrowing from defaulting borrowers.

B = ⁄

µ + ⁄ + ”b1

ˆ T1

0e(R≠µ≠fi)sds + ”

µ + ⁄

⁄

µ + ⁄ + ”b0

ˆ T0

0e(R≠µ≠fi)sds,

and

Bd = ⁄

µ + ⁄ + ”b1e(R≠µ≠fi)T1 + ”

µ + ⁄

⁄

µ + ⁄ + ”b0e(R≠µ≠fi)T0 ,

where T0 = 1R ln

1hb0

a0+hs

2and T1 = 1

R ln1

hb1a0

2.

Equations (15) and (16) are the optimality conditions for the initial borrowing of active and

deactivated lenders when they find their long-term projects. Equation (17) is the condition for

lenders to be willing to roll over the debt. Equation (18) pins down the equilibrium default rate,

which is the ratio of the demand for funding from defaulting borrowers and the total demand of

borrowers. Equation (19) is the market-clearing condition. Equation (20) pins down the portfolio

choice of active lenders.

6.2 The e�ect of collateral risk

As the equilibrium of the full model depends on two key features – collateral risk and maturity

mismatch – let us take one step back before looking at the comparative statics of the full model.

In this subsection, I study the e�ect of collateral risk on liquidity hoarding and cash in the repo

market by studying a state-contingent debt contract with an exogenous borrowing constraint. By

33

shutting down maturity mismatch, we can focus on the e�ect of collateral risk on the equilibrium

outcome. In the next subsection, I will go one step further to characterize the whole equilibrium

with maturity mismatch because of using short-term repo contracts.

6.2.1 Equilibrium with state-contingent repo contracts

The repayment of the repo contract I consider in this subsection is contingent on the arrival of

liquidity when the borrower’s collateral asset or long-term project matures.24 With state-contingent

repo contracts, there are two symmetric equilibria, one in which no agents will default when they

start borrowing and another in which all agents default immediately when they start borrowing.

Proposition 6.2. Suppose that minÓ

f Õ(m0 + a0h ), µy

fl+‰+µhÔ

> (µ+fi)(1+R)fl+‰+µ+fi , then, there exists a

unique stationary equilibrium with the state-contingent repo contract characterized by the following

system of equations,

R = ‰

µ + fi

⁄

fl + ⁄f Õ(m), (21)

m0 ≠ m = ⁄

µ + fi

a0(1 + R)h, (22)

Proof of the proposition is left to the Appendix. The assumption in the proposition ensures

that an agent who borrows to invest in her long-term project would borrow against all her collateral

assets because the return from project investment is high enough. According to proposition 6.2, the

interest rate increases in equilibrium liquidity hoarding of agents waiting for their own investment

opportunities, taking parameter values as given. In the next proposition, I do some comparative

statics.

Proposition 6.3. Under the assumption in Proposition 6.2,

1. ˆmˆ‰ > 0, ˆR

ˆ‰ > 0, ˆiˆ‰ > 0;

2. ˆmˆfi > 0, ˆR

ˆfi < 0, ˆiˆfi > 0.

Proposition 6.3 gives the comparative statics with respect to collateral risk and the maturity

of the project. When collateral risk increases, agents waiting for their investment opportunity24Without loss of generality, I assume that the liquidity at the maturity of either the project or the collateral asset

is enough to repay all the debt.

34

refrain from inter-bank lending and increase storage, as consumptions good in storage would be

the only resource available when the economy is in the bad state. As a result, funding for the

repo market dries up. Thus, the interest rate of the repo contract increases and investment in

long-term investment drops. When the maturity of the project decreases, liquidity from maturing

projects flows more frequently to lenders and, subsequently , results in more project investment and

liquidity hoarding. With more funding available to lenders, the tension between liquidity hoarding

and project investment is tempered. Consequently, both liquidity hoarding and investment increase

and interest rates drop.

6.3 Market liquidity, solvency and balance sheet of financial intermediaries

In this subsection, I examine the comparative statics of the rollover equilibrium, keeping the ex-

pected payo� from dividends from collateral fixed.25 This way, the e�ect of collateral risk on the

rollover equilibrium will be clearer.26

Figure 10 illustrates how collateral risk a�ects the liquidity of the repo market and the solvency

of financial intermediaries. Figure 11 illustrates how collateral risk a�ects the aggregated balance

sheet of financial intermediaries. When the arrival rate of devaluation shock increases, banks hoard

more liquidity in their storage account and refrain from lending to other banks. This increases the

interest rate and haircut in the market, which is consistent with the analytical results we derive

in an environment without maturity mismatch. With maturity mismatch between investment and

liability, borrowers are more likely to default, facing the higher haircut and interest rates. The

increase in default rate would further increase the haircut and interest rates.

The disfunctioning of the repo market is reflected in the aggregate long-term investment and

leverage ratio, as collateral risk, the long-term investment and therefore the aggregate output of the

economy drops. And Figure 11 shows that repo lending contributes less to long-term investment,

and more investment uses funding from storage.

In Figure 11, the increase in storage, which represents liquidity hoarding, and the decrease in

repo lending as collateral risk increases are not completely substitutable. As collateral risk increases,

default is more likely, rollover is harder and lenders have more illiquid collateral asset but less liquid25When I increase devaluation shock ‰, I also increase dividend payment from a collateral tree when the economy

is in the good state. So the dividend payment y satisfies the following condition, yfl+µ+‰ = y0

fl+µ+‰0.

26Parameter values used in the comparative statics and transition dynamics is listed in Table 2 in the appendix.

35

0.01 0.02 0.03 0.040

0.02

0.04

0.06

0.08

0.1

0.12

interest

rate

�0.01 0.02 0.03 0.04

0

10

20

30

40

haircut

(percentage)

�

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.0450

0.02

0.04

0.06

0.08

0.1

default

rate

arrival rate of devaluation shock �

Figure 10: Collateral risk and the liquidity of the repo market and solvency of financial intermedi-

aries.

asset. As a result, the sum of repo lending and storage, which is what we call short-term asset,

decreases. Notice that with the state-contingent repo contract, the sum of repo lending and reserve

is constant. The decrease in reflects the increase in default rate. When debt rollover is harder, the

value of repo lending drops and illiquid collateral asset held by agents waiting for their investment

opportunities increases.

While comparative statics in the model resemble a Kiyotaki-Moore type of model assuming

exogenous borrowing constraint, the constraint in my model is endogenous and depends on the

collateral risk and the contract structure. For example, the haircut for the overnight repo contract

would be higher than that for the state-contingent repo contract. The endogenous haircut is

especially relevant when we study equilibrium dynamics, where the borrowing constraint implied

from haircut will move endogenously along the dynamic path. I now turn to characterize the

transition dynamics triggered by an unexpected shock to the riskiness of collateral assets.

36

0.01 0.02 0.03 0.04

3

3.1

3.2

3.3

3.4

3.5

LT

investm

ent

�0.01 0.02 0.03 0.04

0.14

0.16

0.18

0.2

0.22

0.24

loan

to

collateral

�

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040

1

2

3

4

5

�

repo lending

hoarding

short−term asset

Figure 11: Credit risk and the aggregated balance sheet of the securitized banking system.

7 Equilibrium dynamics

In this section, I characterize the transition dynamics when agents’ expectations about the collateral

risk become more pessimistic. The exercise is intended to capture the transition dynamics triggered

by a shock similar to the one that we experienced during the great recession. More importantly, it

will help us understand the stability of the shadow banking system to shocks to collateral risk.

I assume that the economy before t = 0 is the steady-state equilibrium with Êt = 1, with the

parameter values given in Table 2. At t = 0, ‰ switches from 0.01 to 0.02. The algorithm to

compute the transition dynamics is in the Appendix.

Figure 13 shows the transition dynamics for the liquidity of the repo market and the solvency

of financial intermediaries. Figure 14 illustrates the transition dynamics for the aggregated balance

sheet of financial intermediaries in the securitized banking system. And Figure 15 illustrates the

transition dynamics of output and investment and the impact of the financial crisis on the real

economy.27

27In Figure 14 and Figure 15, I normalized the values of the variables in the initial steady state to one.

37

The stability can be measured by two dimensions: initial response and persistence of the impact.

Initial response of the shadow banking system

Let’s first look at the initial response of the economy to the shift in expectations. At the moment

the expectations shift, haircut increases discontinuously and overshoots to a level even higher than

the haircut in the new steady state. This means that the debt limit that borrowers face given their

collateral asset holding drops discontinuously. As a result, borrowers with debt holdings above the

new debt limit but below the old debt limit are forced to default at t. The mass of default drains

the funding available in the repo market. Thus, the aggregate amount of repo lending drops, and

the massive initial default shows up in the discontinuous drop in repo lending in Figure 14. The

overshooting of haircut increases the probability of the initial default. The figure shows that about

25% of outstanding repo borrowing defaults at the moment the shock hits the economy.

The hike in haircut reflects the fact that the massive default drains liquidity from the market

as lenders’ liquid funding is replaced by illiquid collateral assets from defaulting borrowers. With

less funding available for both liquidity hoarding and repo lending, the interest rate overshoots.

As new entrants bring in more liquidity over time, the interest rates and haircut drops, and

liquidity hoarding increases. But the crisis is not over yet. The counterparty default risk remains

high. The overshooting in the default rate drains the financial system’s liquidity and triggers more

default in the future. This leads us to the persistence of the crisis.

Persistence of the impact of the shock

Massive initial default and high default rate related to the debt overhang for borrowers who started

borrowing before the crisis leads to more default in the future. We can see that the overshooting in

drop in investment, repo borrowing or lending and depressed liquidity hoarding persist for a long

period. If we think of a period as a quarter, it would still take more than ten years for the economy

to recover from the crisis and recover to the output level in the new steady state.

The disruption in the financial system has a big impact on investment and output of the

economy. Figure 15 shows that total investment drops by 20% immediately after the shock to

collateral risk and eventually drops by about 12%. The aggregate output responds with delay

because it takes time for investment in new projects to yield output. Because of the contagion

38

e�ect and the initial massive default, the decrease in output also overshoots. The maximum drop

is about 7%, after which the output recovers a little, and the output at the new steady state is

about 5% lower than at the old steady state.

The fluctuation on the transition path is related to the massive initial default and the out-

standing repo borrowing when the shock hits the economy. Before the shock, investors borrow

more without worrying too much about default because of lower interest rates and lower haircut.

When collateral risk increases unexpectedly, borrowers with outstanding repo borrowing appear

to be over-borrowing and, therefore, likely to default under the higher haircut and interest rate.

The hike in the default rate leads to fluctuations in market liquidity and the balance sheet of the

shadow banking system. Figure 12 illustrates the contagion of counterparty default. It shows two

components of the demand for repo borrowing: one (the blue line) from investors who are still

waiting for their investment opportunities but have illiquid portfolios (more collateral asset and

less consumption goods than their initial endowment) at time zero, another (the red line) from

the rest of the borrowers, including those who started borrowing before time zero and were not

forced to default at time zero, those with liquid portfolio before they start borrowing and those

with illiquid portfolio before start borrowing because they experienced counterparty default after

time zero. The first component first increases and then, declines, with a sharp turn. The turn

takes place when investors whose demand for repo borrowing is included in the first component

start to default. The peak helps us see more clearly the e�ect of intertemporal contagion through

default. Since default triggers more default, default rate for this group of investors remain high.

And because of the massive initial default, the high default rate of this group shows up as a spike

in the aggregate default rate shown in Figure 13. The second component includes those investors

who started borrowing before time zero and were not forced to default at time zero. The demand

from those investor constitutes a debt overhang problem: Since they started borrowing when the

economy was in the initial steady-state equilibrium, in which agents expect that collateral risk is

low, they borrow more than they would have if they had realized the risk; this means the default

rate for this group of borrowers with over-hang debt will be high. Since there are two types of

borrowers with di�erent initial borrowings and debt limits in the steady state before the shock

to expectation hits the economy, the distribution of borrowers is a combination of two continuous

distributions. This is why we can see two distinct peaks in default rate at the beginning of the

39

0 10 20 30 40 50 60 70 80−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

dem

and

time

Figure 12: Decomposition of repo demand

transition path, before investors in the first group start to default.

Overall, the debt overhang and initial massive default lead to fluctuations in default rate and the

hikes in default rate after time zero leads to more default on the dynamic path. The contagion of

counterparty default and illiquidity of lenders’ portfolio leads to persistent overshooting in default

rate, haircut, interest rates, leverage ratio, investment and output.

After the breakout of crisis in 2007, the Federal Reserve stepped in to provide liquidity to the

dealer banksAdrian et al. [2009]. In daily news, many argue whether bailing out “too big to fail”

dealer banks is a good idea. From this exercise, we can see a counterfactual equilibrium dynamics

for what would have happened after the shift in expectation when the Federal Reserve Bank had

not stepped in. The mass of default would trigger over-shooting in haircut and interest rate and

misallocation and fluctuation of lenders’ funding between repo lending and reserve.

40

0 10 20 30 40 50

20

25

30

haircut

h(%

)

time

0 10 20 30 40 500.06

0.08

0.1

0.12

interest

rate

R

time

0 10 20 30 40 50

0.06

0.08

0.1

0.12

0.14

default

rate

time

Figure 13: Equilibrium dynamics of the liquidity of the repo market and solvency of financial

intermediaries, when collateral risk increases unexpectedly.

41

0 10 20 30 40 50

0.16

0.17

0.18

0.19

0.2

0.21

0.22

loan

to

collateral

time

0 10 20 30 40 50

1

1.2

1.4

1.6

1.8

liquidity

hoarding

time

Figure 14: Equilibrium dynamics of the aggregate balance sheet of the securitized banking system,

when collateral risk increases unexpectedly.

42

0 10 20 30 40 50

0.8

0.85

0.9

0.95

1

investm

ent

time

0 10 20 30 40 50

0.92

0.94

0.96

0.98

1

output

time

Figure 15: Equilibrium dynamics of investment and output, when collateral risk increases unex-

pectedly.

43

8 Conclusion

In this paper, I build a dynamic model to study the e�ciency and stability of the shadow banking

system. I show that collateral risk leads to an increase in counterparty default risk in the equilibrium

with repo rollover. Counterparty default drains liquidity from the repo market and reduces output.

By studying the dynamic equilibrium triggered by a shock to collateral risk, I show that the shift

in collateral risk could be an important contributor to the disruptions in the repo market and the

shadow banking system we observed during the 2007-2008 financial crisis. And the shadow banking

system is vulnerable to shifts in market participants’ perception of the collateral risk in two senses.

First, a small shift in the market belief could trigger a massive initial default. Second, the e�ect

of the shock is long-lasting. The exercise shows that the externality of counterparty default has

important implications for the e�ciency and stability of the shadow banking system and related

government policies.

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Thomas Sargent. Liquidity? Keynote address for Bank of Korea Conference, 2013.

Manmohan Singh and James Aitken. The (sizable) role of rehypothecation in the shadow banking

system. International monetary fund (IMF), 2010.

Stephen Williamson and Randall Wright. New monetarist economics: Models. Handbook of Mon-

etary Economics, 3:25–96, 2010.

Stephen D Williamson. Scarce collateral, the term premium, and quantitative easing. 2013.

48

parameter value parameter value

fl 0.03 µ 0.033

⁄ 0.09 fi 0.1

‰ 0.01 y 1

◊ 4 – 0.38

a0 4.17 m0 1

Table 2: Parameter values used in numerical exercises

A Laws of motion

First if Êt = 1, the law of motion of F0t(a, m, s) is

dF 10t(a, m, s) = ≠

ˆI

Q

ccca

aÕ Æ a,

m0t(aÕ, mÕ, sÕ) Æ m,

s0t(aÕ, mÕ, sÕ) Æ s

R

dddb(µ + ⁄) dt(1 ≠ d0t(aÕ, mÕ, sÕ))dF 1

0,t≠dt (aÕ, mÕ, sÕ) (23)

≠ˆ

I

Q

ccca

aÕ Æ a,

mÕ Æ m,

sÕ Æ s

R

dddbd0t(aÕ, mÕ, sÕ)dF 1

0,t≠dt (aÕ, mÕ, sÕ)

+ˆ

I

Q

ccca

aÕ Æ a,

m0t(aÕ, mÕ, sÕ) Æ m,

s0t(aÕ, mÕ, sÕ) Æ s

R

dddb”tIsÕ>0dt (1 ≠ z0t(aÕ, mÕ, sÕ)) dF 1

0,t≠dt(aÕ, mÕ, sÕ)

+ˆ

I

Q

ccca

aÕ + htsÕ Æ a,

m0t(aÕ, mÕ, sÕ) Æ m,

s0t(aÕ, mÕ, sÕ) Æ s

R

dddbd0t(aÕ, mÕ, sÕ)dF 1

0,t≠dt(aÕ, mÕ, sÕ)

+ ÷dtI

Q

ccca

a Æ a0,

m0t(a0, 0, m0) Æ m,

s0t(a0, 0, m0) Æ s

R

dddb

The first component in equation 23 is the outflow of maturing assets and agents receiving an

investment opportunity, conditional on the agent not defaulting. Agents with maturing assets

consumes and leave the economy. Agents with incoming projects flow to the measure F1t. The

second component is the outflow of defaulting agents from type (aÕ, mÕ, sÕ) to other types. The third

component is the inflow of agents who meet defaulting borrowers at t ≠ dt but do not accept them.

49

The fourth component is the inflow from defaulting agents. The last component is the inflow from

newcomers at t ≠ dt.

dF 1dt(a, m) = ≠

ˆI

Q

ccca

aÕ Æ a,

mdt(aÕ, mÕ) Æ m,

sdt(aÕ, mÕ) Æ s

R

dddb(µ + ⁄) dtdF 1

0,t≠dt (aÕ, mÕ) (24)

+ˆ

I

Q

ccca

aÕ + (1 + h)s0t(aÕ, mÕ, sÕ) Æ a,

m0t(aÕ, mÕ, sÕ) Æ m,

s0t(aÕ, mÕ, sÕ) Æ s

R

dddb”tIsÕ>0dtz0t(aÕ, mÕ, sÕ)dF0,t≠dt(aÕ, mÕ, sÕ)

Similarly, if Êt = 1, the law of motion of F1t(a, m, s, i) is

50

dF1t(a, m, s, i) = ≠ˆ

I

Q

cccccca

aÕ Æ a,

m1t(aÕ, mÕ, sÕ) Æ m,

s1t(aÕ, mÕ, sÕ) Æ s,

iÕ Æ i

R

ddddddb(µ + fi) dt(1 ≠ d1t(aÕ, mÕ, sÕ, iÕ))dF1,t≠dt (aÕ, mÕ, sÕ, iÕ)

(25)

≠ˆ

I

Q

cccccca

aÕ Æ a,

mÕ Æ m,

sÕ Æ s,

iÕ Æ i

R

ddddddbd1t(aÕ, mÕ, sÕ, iÕ)dF1,t≠dt (aÕ, mÕ, sÕ, iÕ)

+ˆ

I

Q

cccccca

aÕ + hts1t(aÕ, mÕ, sÕ, iÕ) Æ a,

m1t(aÕ, mÕ, sÕ, iÕ) Æ m,

s1t(aÕ, mÕ, sÕ, iÕ) Æ s,

iÕ Æ i

R

ddddddb”tIsÕ>0dtz1t(aÕ, mÕ, sÕ, iÕ)dF1,t≠dt(aÕ, mÕ, sÕ, iÕ)

+ˆ

I

Q

cccccca

aÕ Æ a,

m1t(aÕ, mÕ, sÕ) Æ m,

s1t(aÕ, mÕ, sÕ) Æ s,

iÕ Æ i

R

ddddddb”tIsÕ>0dt (1 ≠ z1t(aÕ, mÕ, sÕ, iÕ)) dF1,t≠dt(aÕ, mÕ, sÕ, iÕ)

+ˆ

I

Q

cccccca

aÕ + (1 + h)sÕ Æ a,

m1t(aÕ, mÕ, sÕ, iÕ) Æ m,

s1t(aÕ, mÕ, sÕ, iÕ) Æ s,

iÕ Æ i

R

ddddddbd1t(aÕ, mÕ, sÕ, iÕ)dF1,t≠dt(aÕ, mÕ, sÕ, iÕ)

+ ⁄dt

ˆI

Q

cccccca

aÕ Æ a,

iIt(aÕ, mÕ, sÕ) Æ i,

mIt(aÕ, mÕ, sÕ) Æ m,

sIt(aÕ, mÕ, sÕ) Æ s

R

ddddddbdF0,t≠dt(aÕ, mÕ, sÕ)

+ ⁄dt

ˆI

Q

cccccca

aÕ Æ a,

iIt(aÕ, mÕ, 0) Æ i,

mIt(aÕ, mÕ, 0) Æ m,

sIt(aÕ, mÕ, 0) Æ s

R

ddddddbdFd,t≠dt(aÕ, mÕ) (26)

If Êt≠dt = 0, F 10t(Œ, Œ, Œ) = 0, F 1

dt(Œ, Œ) = 0,F 11t(Œ, Œ, Œ, Œ) = 0 , and

51

dG0t(m) = ≠ (µ + ⁄) dt

ˆ{m(mÕ)Æm}

dG0t≠dt(mÕ) + ÷dtI;m0A Æ m

< (27)

dG1t(m, i) = ≠fidt

ˆ{mI0(mÕ)Æm}

dG1,t≠dt(mÕ, i) + ⁄dt

ˆIY_]_[

iI0(mÕ) Æ i,

mI0(mÕ) Æ m

Z_

_\

dG0t≠dt(mÕ) (28)

If Êt≠dt = 1, then conditional on the Êt = 0,

G0t(m) =ˆ

I;mÕ Æ m

< [1 ≠ (µ + ⁄) dt] dF0,t≠dt!aÕ, mÕ, sÕ" (29)

ˆI;

mÕ Æ m

< [1 ≠ (µ + ⁄) dt] dFd,t≠dt!aÕ, mÕ"

+ ÷dtI;m0A(w) Æ m

<

G1t(m, i) =ˆ

IY_]_[

mÕ Æ m,

iÕ Æ i

Z_

_\

[1 ≠ (µ + fi) dt] dF1,t≠dt!aÕ, mÕ, sÕ, iÕ" (30)

+ ⁄dt

ˆIY_]_[

m0A(mÕ) Æ m,

i0A(mÕ) Æ i

Z_

_\

dF0,t≠dt(aÕ, mÕ, sÕ)

+ ⁄dt

ˆIY_]_[

m0A(mÕ) Æ m,

i0A(mÕ) Æ i

Z_

_\

dFd,t≠dt(aÕ, mÕ)

B E�ciency

Proof for Proposition 5.1:

52

Proof.

St = max{C· ,m· ,i· }’·Øt

ˆ Œ

tC· e≠fl(·≠t)d·,

s.t.C· d· + (⁄d·) ÷

µ+⁄ i· + m· Æ´ ·

≠Œfl+fi

fi ◊i–s (fid·) e≠fi(·≠s)⁄ ÷

µ+⁄ds e≠fl(·≠t)“·

+´ ·

≠Œ ayÊ· (µd·) e≠µ(·≠s)÷ds + (÷d·) M· + m·≠d· ,

C· , m· , i· Ø 0. “c· , “l· , “m· , “i·

FOC

C· : e≠fl(·≠t)d· ≠ e≠fl(·≠t)“· d· + “c· = 0

m· : ≠e≠fl(·≠t)“· + e≠fl(·+d·≠t)“·+d· + “m· = 0

i· : ≠⁄d· ÷µ+⁄e≠fl(·≠t)“· +

´Œ· ◊–i–≠1

· (fl + fi) d·e≠fi(s≠·)⁄ ÷µ+⁄e≠fl(s≠t)“sds + “i· = 0

≠ 1 + ◊–i–≠1·

ˆ Œ

·(fl + fi) e≠(fl+fi)(s≠·)ds = 0

◊–i–≠1· = 1

Substituting the resource constraint to the objective function ...

Ct ≠ ÷mt + ⁄÷

µ + ⁄it =

ˆ ·

≠Œ

5fl + fi

fi◊i–

s fie≠fi(·≠s)6

⁄÷

µ + ⁄ds +

ˆ ·

≠Œaµe≠µ(·≠s)÷ds

= fl + fi

fi◊i–

s ⁄÷

µ + ⁄+ a÷

Cú ≠ ÷mú =5

fl + fi

fi◊iú– ≠ iú

6⁄÷

µ + ⁄+ a÷

C the e�ect of collateral risk

C.1 Equilibrium with state-contingent collateralized debt contract and exoge-

nous haircut

The contract is contingent on liquidity arrival: At the date of borrowing, the borrower puts down h

units of collateral for each unit of consumption good she borrows; when the borrower has liquidity

53

to repay the debt, the repayment is 1 + R, and if the borrower does not repay the debt, the lender

will keep the collateral.

The borrower’s expected payo� when she invests in a long-term project is

U(a0, m, s) = maxb

f(m + s + b) + µy

fl + ‰ + µa0 ≠ (µ + fi)(1 + R)b

fl + ‰ + µ + fi

s.t.0 Æ b Æ a0(1 + R)h

FOC:

f Õ(m + s + b) ≠ “1 ≠ “0 ≠ (µ + fi) (1 + R)fl + ‰ + µ + fi

= 0

If the optimal borrowing is such that f Õ(m + s + b) > (µ+fi)(1+R)fl+‰+µ+fi , then b = a0

(1+R)h . Otherwise,

f Õ(m + s + b) = (µ+fi)(1+R)fl+‰+µ+fi .

Assume that ◊ is high enough and a0 and m0 are small enough so that, f Õ(m+s+b) > (µ+fi)(1+R)fl+‰+µ+fi ,

or f Õ(m + s + a0h ) > (µ+fi)(1+R)

fl+‰+µ+fi , then,

U(a0, m, s) = f3

m + s + a0h

4+

5µy

fl + ‰ + µ≠ µ + fi

(fl + ‰ + µ + fi) h

6a0

The value function of an agent before she finds her long-term project is

(fl + µ + ‰ + ⁄)V (a0, m, s) = µ(a0y + m0) + ‰⁄

fl + ⁄f(m) + (µ + fi)Rs

+ ⁄;

f3

m + s + a0h

4+

5µy

fl + ‰ + µ≠ µ + fi

(fl + ‰ + µ + fi) h

6a0

<

(fl + µ + ‰ + ⁄)V (a0, m, m0 ≠ m) = µ(a0y + m0) + ‰⁄

fl + ⁄f(m) + (µ + fi)R (m0 ≠ m)

+ ⁄;

f3

m0 + a0h

4+

5µy

fl + ‰ + µ≠ (µ + fi)(1 + R)

(fl + ‰ + µ + fi) h

6a0

<

The optimal choice of portfolio implies that dV (a0,m,m0≠m)dm = 0.

R = ‰

µ + fi

⁄

fl + ⁄f Õ(m)

54

The equilibrium is solved by the following system of equations:

R = ‰

µ + fi

⁄

fl + ⁄f Õ(m),

m0 ≠ m

µ + ⁄= ⁄

µ + ⁄

a0(1 + R)h

1µ + fi

,

the last equation being the market-clearing condition.

With the system of equations, the net supply to the market for the state-contingent contract,

�, can be reduced to a function of storage holding m.

(µ + ⁄)�(m) = m0 ≠ m ≠ ⁄

µ + fi

a0(1 + R)h

= m0 ≠ m ≠ ⁄

µ + fi

a0h

11 + ‰

µ+fi⁄

fl+⁄f Õ(m)

�(0) = 0 and (µ + ⁄)�(m0) = ≠ ⁄µ+⁄

a0h

11+ ‰

µ+fi⁄

fl+⁄ f Õ(m0) < 0.

(µ + ⁄)�Õ(m) = ≠1 + ⁄

µ + fi

a0h

‰µ+fi

⁄fl+⁄f ÕÕ(m)

Ó1 + ‰

µ+fi⁄

fl+⁄f Õ(m)Ô2 < 0

So, if the equilibrium exists, it is unique.

Comparative statics

(µ + ⁄)ˆ�ˆ‰

= ⁄

µ + fi

a0h

1µ+fi

⁄fl+⁄f Õ(m)

Ë1 + ‰

µ+fi⁄

fl+⁄f Õ(m)È2 > 0

(µ + ⁄)ˆ�ˆfi

= ≠ ⁄

µ + fi

a0h

11 + ‰

µ+fi⁄

fl+⁄f Õ(m)

Y]

[≠ 1µ + fi

≠≠ ‰

(µ+fi)2⁄

fl+⁄f Õ(m)

1 + ‰µ+fi

⁄fl+⁄f Õ(m)

Z^

\

= ⁄

(µ + fi)2a0h

11 + ‰

µ+fi⁄

fl+⁄f Õ(m)1

1 + ‰µ+fi

⁄fl+⁄f Õ(m)

> 0

55

D Characterization of equilibrium with debt rollover

D.1 The problem of agents with a long-term project

Given the price of assets at fire sale, implied by h, the maximum amount of borrowing is a0h . At the

moment of default, the borrower’s continuation value is f(i) + µfl+µ+‰(a ≠ a0). For a borrower with

a assets decides to default when she puts down a0 units of asset as collateral, the value function

can be solved by the following di�erential equation. It is easy to verify that an agent will hold zero

storage after her long-term investment. So, we denote W (a, b, 0, i) = max0Æa0Æa W (a, b, a0, i)

flW (a, b, a0, i) = fi5

fl + fi

fif(i) + µ

fl + µ + ‰ya ≠ b ≠ W (a, b, a0, i)

6

+ ‰ [f(i) ≠ W (a, b, a0, i)]

+ µ [f(i) + ya ≠ b ≠ W (a, b, a0, i)] + ˆW (a, b, a0, i)ˆb

bR,

W3

a,a0h

, a0, i4

= f(i) + µy

fl + µ + ‰(a ≠ a0),

where W1a, a0

1+h , a0, i2

is the continuation when she defaults.

Lemma D.1.

W (a, b, a0, i) =5

µ + fi ≠ ‰·

fl + ‰ + µ + fi ≠ R

1h

≠ µ

fl + µ + ‰y

6a0

3hb

a0

4(fl+‰+µ+fi)/R

≠ µ + fi ≠ ‰·

fl + ‰ + µ + fi ≠ Rb + f(i) + µ

fl + µ + ‰ya

Proof.

(fl + ‰ + µ + fi) W (a, b, a0, i) = (fl + ‰ + µ + fi) f(i) ≠ (µ + fi)b + µfl + µ + ‰ + fi

fl + µ + ‰ya + ˆW (a, b, a0, I)

ˆbbR

Guess W (a, b, a0, i) = C0b(fl+‰+µ+fi)/R + C1b + C21f(i) + µ

fl+µ+‰ya2

+ C3. Then ˆW (a,b,a0,I)ˆb =

fl+‰+µ+fiR C0b(fl+‰+µ+fi≠R)/R + C1.

(fl + ‰ + µ + fi)5C0b(fl+‰+µ+fi)/R + C1b + C2

3f(I) + µ

fl + µ + ‰ya

4+ C3

6

=(fl + ‰ + µ + fi)f(I) ≠ (µ + fi ≠ ‰·)b + µfl + µ + ‰ + fi

fl + µ + ‰ya

+ bR5

fl + ‰ + µ + fi

RC0b(fl+‰+µ+fi≠R)/R + C1

6

56

(fl + ‰ + µ + fi) C1 = RC1 ≠ (µ + fi ≠ ‰·)

(fl + ‰ + µ + fi) C2 = (fl + ‰ + µ + fi)

(fl + ‰ + µ + fi) C3 = 0

C1 = ≠ µ+fi≠‰·fl+‰+µ+fi≠R , C2 = 1, C3 = 0.

Therefore, W (a, b, a0, I) = C0b(fl+‰+µ+fi)/R≠ µ+fi≠‰·fl+‰+µ+fi≠Rb+f(i)+ µ

fl+µ+‰ya. Since W1a, a0

1+h , a0, i2

=

f(i) + µyfl+µ+‰(a ≠ a0), we have

C0

3a0h

4(fl+‰+µ+fi)/R

≠ µ + fi

fl + ‰ + µ + fi ≠ R

a0h

+ f(i) + µ

fl + µ + ‰ya = f(i) + µy

fl + µ + ‰(a ≠ a0)

C0 =5

µ + fi

fl + ‰ + µ + fi ≠ R

1h

≠ µ

fl + µ + ‰y

6a0

3a0h

4≠(fl+‰+µ+fi)/R

W (a, b, a0, i) =5

µ + fi

fl + ‰ + µ + fi ≠ R

1h

≠ µ

fl + µ + ‰y

6a0

3hb

a0

4(fl+‰+µ+fi)/R

≠ µ + fi

fl + ‰ + µ + fi ≠ Rb + f(i) + µ

fl + µ + ‰ya

Lemma D.2. The partial derivative of the value function is

ˆW (a, b, a0, i)ˆy

= µ

fl + µ + ‰

C

a ≠ a0

3hb

a0

4(fl+‰+µ+fi)/RD

ˆW (a, b, a0, i)ˆa0

=5

µ

fl + µ + ‰y ≠ µ + fi

fl + ‰ + µ + fi ≠ R

1h

6fl + ‰ + µ + fi ≠ R

R

3hb

a0

4(fl+‰+µ+fi)/R

ˆW (a, b, a0, i)ˆa

= µ

fl + µ + ‰y

ˆW (a, b, a0, i)ˆb

=5

µ

fl + µ + ‰yh ≠ µ + fi

fl + ‰ + µ + fi ≠ R

6fl + ‰ + µ + fi

R

C

≠3

hb

a0

4(fl+‰+µ+fi≠R)/RD

≠ µ + fi

fl + ‰ + µ + fi ≠ R

ˆW (a, b, a0, i)ˆi

= f Õ(i)

From Lemma D.2, we can see that the borrower will roll over her debt up to the debt limit as

long as µfl+µ+‰y ≠ µ+fi

fl+‰+µ+fi≠R1h > 0. And, in this case, W (a, ≠b, i) = W (a, b, a, i).

57

Lemma D.3. agents

• will borrow against all their asset if µfl+µ+‰y ≠ µ+fi

fl+‰+µ+fi≠R1h > 0 or h > µ+fi

fl+‰+µ+fi≠Rfl+µ+‰

µy .

• will not borrow if h < µ+fifl+‰+µ+fi≠R

fl+µ+‰µy

• are indi�erent between borrowing or not if h = µ+fifl+‰+µ+fi≠R

fl+µ+‰µy .

D.2 The problem of agents when they find a long-term project

Next, I consider the optimization problem at the moment the agent finds an investment opportunity.

U(a, m, s) = maxcØ0,0ÆbÆ a0

1+h

c + W (a, ≠b, s + m ≠ c + b)

The first-order condition of the problem isdW (a, ≠b, s + m ≠ c + b)

db= 0,

ˆ

ˆbW + ˆ

ˆiW = 0.

From the first-order condition we have the following lemma.

Lemma D.4. The optimal choice of project investment and initial borrowing of an agent with

portfolio (a, s, m) is solved by equation

3hb

a

4(fl+‰+µ+fi≠R)/R

= R

fl + ‰ + µ + fi

f Õ(s + b) ≠ µ+fifl+‰+µ+fi≠R

µfl+µ+‰yh ≠ µ+fi

fl+‰+µ+fi≠R

Let b = hba+hsÕ . b = a+hsÕ

h b, s ≠ sÕ + b = s + ah b ≠ (1 ≠ b)sÕ. Let

� =

˵

fl+µ+‰yh ≠ µ+fifl+‰+µ+fi≠R

Èfl+‰+µ+fi

R b(fl+‰+µ+fi≠R)/R

≠Ëf Õ

1s + a

h b ≠ (1 ≠ b)sÕ2

≠ µ+fifl+‰+µ+fi≠R

È

.

ˆ�ˆsÕ = (1 ≠ b)f ÕÕ

3s + a

hb ≠ (1 ≠ b)sÕ

4< 0

ˆ�ˆb

=5

µ

fl + µ + ‰yh ≠ µ + fi

fl + ‰ + µ + fi ≠ R

6fl + ‰ + µ + fi

R

fl + ‰ + µ + fi ≠ R

Rb(fl+‰+µ+fi≠2R)/R

≠3

a

h+ sÕ

4f ÕÕ

3s + a

hb ≠ (1 ≠ b)sÕ

4> 0

Then, from the Implicit Function Theorem, we have the following result:

58

Lemma D.5. Counterparty default increases the default probability of lenders in the future

db

dsÕ > 0

Since default probability is an increasing function of b, we know from this lemma that counter-

party default that transforms liquid funding to collateral assets will increase the default probability

when the agent starts borrowing.

D.3 The problem of agents waiting for an investment opportunity

In the stationary environment, the value functions of agents waiting for their investment opportu-

nities, given the optimal portfolio choice on storage m and lending s, can be written as follows:

(fl + µ + ‰ + ⁄ + ”) V (a, m, s) = sR + µ (ya + s + m) + ”V d (a + hs, m) + ‰V A(m) + ⁄U(a, m, s),

(fl + µ + ‰ + ⁄) V d (a + hs, m) = µ [y (a + hs) + m] + ‰V A(m) + ⁄U(a + hs, m, 0).

(fl + ⁄)V A(m) = ⁄◊m–

Lemma D.6. The following condition must be satisfied in equilibrium:

µ (yh ≠ 1) s =sR + ⁄ [U(a, s, m) ≠ U(a + hs, 0, m)]

Proof. Suppose, instead, that V d(a + hs, m) > V (a, m, s). Then lenders will default on the loan

by keeping the collateral. So, V d(a + hs, m) Æ V d(a, m, s). Suppose, V d(a + hs, m) < V (a, m, s).

Then, lenders won’t be willing to lend to the defaulting borrowers when they observe that the

borrowers are going to default. Rollover is not possible.

Given the value functions, the optimal portfolio choice is solved by the following problem:

maxs,mœR+

V (a, m, s)

s.t.s + m Æ m0

59

E Equilibrium dynamics with a constant ‰

Given the initial condition, the distribution of lenders and borrowers, we need to solve the sequence

of default rate”t, interest rate Rt, haircut ht, and the optimal portfolio choice of active lenders

between repo lending st and storage mt.

Active lenders’ portfolio choice

Given the value functions,

Vt(a, m, s) = maxc,m,s

sRtdt + c + µdt(ay + m + s) + ”tdte≠fldtVt+dt (a + hts, m, 0)

+‰dt ⁄fl+⁄f(m) + ⁄dtWt(a, m, s)

+ [1 ≠ (µ + ”t + ‰ + ⁄) dt] e≠fldtVt+dt(a, m, s)

s.t.c + m + s Æ m + s

c, m, s Ø 0

We focus on the parameter space where consumption allocation before agents exit the market

is always 0. Thus, c = 0, and m + s = m + s = m0, Since m0 is a constant, ˆVt(a,m,s)ˆs = Rtdt,

Ut(a, m, m0 ≠ m) does not depend on m given w, and m is solved by the following problem:

max0ÆmÆm0

”tdte≠fldtVt+dt (a + ht(m0 ≠ m), m, 0) + ‰dt⁄

fl + ⁄f(m) ≠ Rtdtm

FOC

”t

1≠ht

ˆˆa + ˆ

ˆm

2Vt+dt (a + ht(m0 ≠ m), m, 0) + ‰ ⁄

fl+⁄f Õ(m) ≠ Rt + “mØ0 ≠ “mÆw = 0

ˆ

ˆaV d

t+dt (a + ht(m0 ≠ m), m) =ˆ Œ

0

5µy + ⁄

d

daUt+s(aÕ, mÕ, 0)

6e≠(µ+⁄+‰+fl)sds

d

daUt(a, m, 0) =

ˆ Tt(b)

0

5µy + fi

µy

fl + ‰ + µ

6e≠(‰+fi+µ+fl)· d·

= µy

fl + ‰ + µ

Ë1 ≠ e≠(fl+fi+µ+‰)Tt(b)

È

60

ˆ

ˆaV d

t+dt (a + ht(m0 ≠ m), m)

=ˆ Œ

0

5µy + ⁄

µy

fl + ‰ + µ

11 ≠ e≠(fl+fi+µ+‰)Tt+s(bt(a+(1+ht)(w≠m),m))

26e≠(µ+⁄+‰+fl)sds

ˆ

ˆmV d

t+dt (a + ht(m0 ≠ m), m)

=ˆ Œ

0

5µ + ⁄

ˆ

ˆmUt+s(a + ht(w ≠ m), m, 0) + ‰

⁄

fl + ⁄f Õ(m)

6e≠(µ+⁄+‰+fl)sds

=µ + ‰ ⁄

fl+⁄f Õ(m)µ + ⁄ + ‰ + fl

+ ⁄

ˆ Œ

0f Õ (m + bt+s (a + ht(w ≠ m), m)) e≠(µ+⁄+‰+fl)sds

Haircut and indi�erence condition

V dt (a, m) =

ˆ Œ

0

5µ(ay + m) + ⁄Ut+s(a, m, 0) + ‰

⁄

fl + ⁄f(m)

6e≠(µ+⁄+‰+fl)sds

=µ(ay + m) + ‰ ⁄

fl+⁄f(m)µ + ⁄ + ‰ + fl

+ˆ Œ

0⁄Ut+s(a, m, 0)e≠(µ+⁄+‰+fl)sds

Vt(a, mt, st) =ˆ Œ

0

S

WUst+· Rt+· + ”t+· V d

t+· (a + ht+· st+· , mt+· ) + µ(ay + m0)

+‰ ⁄fl+⁄f(mt+· ) + ⁄Ut+· (a, m0)

T

XV

e≠´ ·

0 ”t+udu≠(µ+‰+⁄+fl)· d·

Haircut must be such that, given the optimal choice (mt, st),

Vt(a, mt, st) = V dt (a + htst, mt)

Initial borrowing of investors with LT projects

Ut(a, m, s) = max0ÆbÆ a

ht

Rtsdt + f (m + s + b)

+´ Tt(b)

0

ˉ · 0 + µ

1ay ≠ be

´ ·0 Rt+sds

2+ fi

1µay

fl+µ+‰ ≠ be´ ·

0 Rt+sds2È

e≠(‰+µ+fi+fl)· d·

where Tt(b) = infÓ

· : be´ ·

0 Rt+sdsht+· = aÔ

. The maximization problem is equivalent to the follow-

ing problem:

61

max0ÆbÆ a

1+ht

f (m + s + b) +´ Tt(b)

0fl+µ+‰+fi

fl+µ+‰ µaye≠(‰+µ+fi+fl)· d·

≠b´ Tt(b)

0 (µ + fi) e´ ·

0 Rt+sdse≠(‰+µ+fi+fl)· d·

A su�cient condition for equation be´ ·

0 Rt+sds = aht+·

to have at most one solution is that

e´ ·

0 Rt+sdsh· is monotonically increasing. e´ ·

0 Rt+sdsËRt+· ht+· + ht+·

È> 0.

ˆTt(b)ˆb

= ≠ ht+·

bËRt+· ht+· + ht+·

È

Market-clearing condition

The density of deactivated lenders with portfolio (a + s· (1 + h· ), m· , 0) is ”· n0· e≠(µ+⁄)(t≠·). The

measure of active lenders at t is denoted n0t .

Demandt =ˆ Œ

0

ˆ Œ

0⁄”t≠·≠sn1

t≠·≠se≠(µ+⁄)sbt≠·≠s (a + st≠·≠sht≠·≠s, mt≠·≠s, 0) e(R≠fi≠µ)·

I {· Æ Tt≠· (bt≠·≠s (a + st≠·≠sht≠·≠s, mt≠·≠s, 0))} dsd·

+ˆ Œ

0⁄n1

t≠· bt≠· (a, m0, 0) e(R≠fi≠µ)· I {· Æ Tt≠· (bt≠· (a, m0, 0))} d·

Supplyt = n1t st

Default rate

Updating ”t depends on the measure of demand from borrowers and the flow of demand from

defaulting borrowers.

”t = Demand of defaulting borrowert

Demandt

The distribution of the lender’s portfolio, F (a, m, s), a function of timing of counterparty default.

n1t =´Œ

0 ÷e≠(µ+⁄)·≠´ ·

0 ”t≠sdsd· . The measure of agents in the default cohort · (· < t) at moment

t is ”· n1· e≠(µ+⁄)(t≠·). And agents in the default cohort · have portfolio, (a + s· (1 + h· ), m· , 0).

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Numerical algorithm to compute the transition path

Suppose that the shock to expectation arrives at t = 0.

1. Guess a sequence of interest rate, haircut and default rate, {Rt, ht, ”t} ’tØ0.

2. Given the sequence, {Rt, ht, ”t} ’tØ0, solve for the policy functions and value functions of

agents on the transition path.

3. Given the h0, solve for the mass of initial default. And then, given the policy function of

agents, solve the distribution of agents along the transition path.

4. Given the distributions of agents and the policy functions, update the default rate of borrow-

ers.

5. Given the distributions of agents and the policy functions, solve for the net demand of repo

borrowing. Update the interest rate according to the net demand of repo borrowing.

6. Given the value functions of agents, update haircut.

7. Go back to step 2 with the updated sequence of interest rate, haircut and default rate, until

convergence.

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